Stats

Question 1

What are the two kinds of statistics in respect to their use?

Answer 1

1) Descriptive statistics: Measures of central tendency and variability
2) Inferential statistics: Parameter estimation, defining uncertainty, determining reasons for variation.

Question 2

Bias

Answer 2

Any systematic deviation between sample estimates and a true value

Question 3

Inference

Answer 3

Drawing a conclusion from a premise.

Question 4

Premise

Answer 4

A premise is a statement we assume is true (e.g. data and observations).

Question 5

The two kinds of variability in a study

Answer 5

1) Variability related to the variables we’re investigating.
2) Variability that is not interesting in the context of what we are investigating (noise variability).

Question 6

What is the purpose of inferential statistics?

Answer 6

1) To discriminate between interesting variation and noise variation.
2) To determine the probability of observing such variability if a scientific mechanism was not operating.

Question 7

What is an informal way think of “statistically significant” as?

Answer 7

Statistically significant = unlikely to hve ocurred by chance.

Question 8

How does statistical analysis fit into the scientific method?

Answer 8

Statistical analysis allows for an objctive assessment of evidence in support or against a hypothesis.

Question 9

What is a scientific hypothesis?

Answer 9

A scientific hypothesis is a proposed cause and effect relationship between a process and an observation.

Observation = what
Hypothesis = how

Question 10

What is a statiscial hypothesis?

Answer 10

Simply a statment about whether there is or not a pattern of interest in the data.

Question 11

What are the two types of statistical hypotheses?

Answer 11

• H₀* (null hypothesis) = No effect on predictor variable
• H_A* (alternative hypothesis) = Effect on predictor variable

Question 12

What are the two kinds of variables in an experiment?

Answer 12

1) Predictor variable (aka independent variable)
2) Response variable (aka dependent variable)

Question 13

What is µ₀ “mew not” in a one-sample study?

Answer 13

µ₀ is the true population mean.

Question 14

What is α?

Answer 14

α is a set proability criterion we use to reject a null hypothesis. It’s a set chance for incorreclty rejecting a null hypothesis.

Question 15

In testing a hypothesis, what is a sample used for?

Answer 15

In testing a hypothesis, we use a sample to estimate characteristics of an underlying population.

Question 16

The statement “We calculate the proability H₀ is true, given the data” is wrong.

1) Why is this?
2) What is the correct statment?

Answer 16

1) Population paremeters are fixed, so either H₀ is true or not.
2) The correct statment would be “We calculate the probability of oberving the data we gathered given a H₀”.

Question 17

How are the N₀ and N_A formualted in a one-sample test?

Answer 17

• H₀*: µ = µ₀
• H_A*: µ ≠ µ₀

OR

• H₀*: µ - µ₀ = 0
• H_A*: µ - µ₀ ≠ 0

Question 18

To test a hypothesis we use a test statistic. Broadly, how is a test statistic calculated?

Answer 18

Test statistic = effect (i.e. µ - µ₀) / error

Question 19

How is a test statistic used for testing a hypothesis?

Answer 19

1) Either comparing the test statistic to a critical value

or
2) calculating a p-value associated with that test statistic

Question 20

How is a p-value interpreted?

Answer 20

The p-value can e though of as the probability of observing the data if the H₀ was true.

Question 21

In the example of a z statistic, what is z?

Answer 21

z is the number of stander deviations by which the observed mean differs from the population mean.

Question 22

What does the central limit theorem state?

Answer 22

The CLT states that the distribution of means from a non-normal populaion will not be normal but will approximate normalityas n increases.

Question 23

How is population variance calculated?

Answer 23

Question 24

How is sample variance calculated?

Answer 24

Question 25

What is standard error and how is it calculated?

Answer 25

The standar error (aka SE, SEM) is the standard deviation of a statistic (in this case mean) and is calculated as:

Question 26

Noting that we don’t know σ,
How is SE estimated?

Up to 19:00

Answer 26

We can estimate SE as:

This is because the best estimate for population variance σ is sample variance s.

Question 27

What is the relationship between sample size n and variance in the distribution of sample means?

Answer 27

The variance in the distribution of means will decrease as n increases.

Question 28

How si the t statistic calculated

Answer 28

Question 29

1) What is the distribution shape difference between z-distribution and t-distribution?
2) What effect does this have on a critical value?

Answer 29

1) In the t-distribution, there is more area at the distribution tails. Also the t-dsitribution is “pushed at the top”.
2) a t-critical value is more extreme than a z-critical value (see bars in the figure)

Question 30

Note: Remember than for a normal distribution, the percentage of values in an area can be known with the number of standard deviations form the mean.

Answer 30

Question 31

★ one sample t-test example

We want to know whether drug A significantly changes the body temperature of healthy human adults 2 hours after taking the drug. Note that the normal body temperature is 37 °C. We take our measurements from a sample and find a mean temperature of 38.5 °C and a variance of 3.4. The sampe size is 30.

Note: On the final exam we’ll have to calculate variance, which will not be given to us.

Answer 31

s = sqrt(3.4/30) = 0.3366502

t = (38.5 - 37)/0.3366502 = 4.456

Then we look up the t critical value in a table using:
two tailed, 29 df, and an α of 5%,
and get a value of 2.045.

Because 4.456 > 2.045 we reject the null hypothesis and cunclude that drug A significantly changes body temperature.

Question 32

When do we use a t statistic instead of a z statistic?

Answer 32

We can’t use z if we are estimating σ from s.

Question 33

What is the value of v (derees of freedom) for an hypothesis about mean?

Answer 33

v = n - 1

Question 34

How does the location of the critical value of a one-taled test differ from the critical calue of a two-tailed test?

Answer 34

For a one-tailed test, we put the entire rejection region into one tail of the t-distribution, instead of splitting it between the two tails.

Question 35

In the following t-distribution graph, you wold reject the null hypothesis if the t-value was less than the critical value (shown in red).

Question 36

What is t thought of as?

Answer 36

Like z, t is the number of standard deviations from the mean.

Question 37

What are the typed of errors in hypothesis testing?

Answer 37

1) Type I error (α) = rejecting true H₀
2) Type II eror (β) = failing to reject a false H₀

Question 38

Note that: When µ ≠ µ₀, the critical value defines the boundary between power and type II error.

Question 39

In a t-distribution, why do we need to know the degrees of freedom?

Answer 39

The degrees of freedom are needed because the distribution shape changes for different degrees of freedom.

Question 40

Note that t-tables only tell us whether the p-value is greater or less than a specified α.

If instead of using tables, you want to know the p-value, how do you calculate it?

Answer 40

In R, you can use:

1 - pt(4.456,29)

Question 41

In an example similar to the drug A and temperature example, when would you use a one-sampled test?

Answer 41

You would use a one-tail test if you’re only interested in whether body temeprature is either increasing or decreasing as a result fo the drug.

Question 42

In an the drug A and temperature example, how would you write the one-tailed statistical hypotheses in the following cases?

1) We want to know whether the drug increases body temperature
2) We want to know whethr the drug decreases body temperature

Answer 42

1) We want to know whether the drug increases body temperature

• H₀*: µ - µ₀ ≤ 0
• H_A*: µ - µ₀ > 0

2) We want to know whethr the drug decreases body temperature

• H₀*: µ - µ₀ ≥ 0
• H_A*: µ - µ₀

Question 43

Note that the = sign always is part of H₀ and not *H_A

H₀*
: µ - µ₀ ≤ 0
H_A: µ - µ₀ > 0

Question 44

What is the formula for a two-sample t-test looking for any diffeence between the two samples?

Answer 44

Question 45

What is the formula for a two-sample t-test with a given µ0 different than 0 (i.e. looking for a specific difference between two sample means)?

Answer 45

Question 46

In a one-sample t-test we use s to estimate σ.

In a two-sample t-test we do something similar. We assume that s₁ and s₂ are similar, but not the same, so we use s²_p as a pooled variance estimator (see formula).

How is s²_p calculated?

Answer 46

where SS is sum of squares.

Question 47

How is the formula for two-sample one-tailed t-test different from the two-tailed formula?

Answer 47

In the formula for one-tailed t-tests, the values of the means are not absolute.

Question 48

Using the visual representation of a t-distribution, explain why we always need to accept some level of error.

Answer 48

We need to accept some level of error because the t-distribution asymptotes at the x axis and there is no value of t that corresponds to a proability of 0%.

Question 49

What is statistical power?

Answer 49

Statistical power (1 - β) is the probability of correclty rejecting a false H₀

Question 50

What is the relationship between power and the difference between µ and µ₀?

Answer 50

The greater the difference between µ and µ₀, the greater the power we have to deect the difference.

Question 51

What does the probability of a type II error depend on?

Answer 51

The probability of a type II error depends on:

1) what H_A is
2) how large an effect we hope to detect
3) sample size
4) how good the experimental design was

Question 52

When we set an α of 0.05, we often have a β of around 0.2 and a power of around 0.8.

Question 53

★ Welsch’s test example

We want to test for a difference in protein concentration between two pea populations. We determine variances are heterogenous and thus use a Welsch’s test:

Results:
mean_fert = 24 g protein
SS_fert = 261 g2
n_fert = 30

mean_unfert = 21.8 g protein
SS_unfert = 320 g2
n_unfert = 29

Answer 53

• *s²_f** = SS_f / (n_f - 1) = 261/29 = 9
• *s²_u** = SS_f / (n_{<span>u</span>} - 1) = 320/28 = 11.43

t’ = (x̄₁ - x̄₂)/sqrt(s²₁/n₁ + s²₂/n₂)
= (24 - 21.8)/sqrt(9/30 + 11.43/29) = 2.6406

Wilsch has a different distribution, so we need to use a special formula to calculate the degrees of freedom:
v’ = (s²_x̄1 + s²_x̄2)²/(s²_x̄1)²/(n₁ - 1) + (s²_x̄2)²/(n2 - 1)
but first:

• *s²_x̄f** = s²_f/n_f = 9/30 = 0.3
• *s²_x̄u** = s²_u/n_{<span>u</span>} = 11.43/29 = 0.3941
• *v’** = (s²_x̄1 + s²_x̄2)²/[(s²_x̄1)²/(n₁ - 1) + (s²_x̄2)²/(n2 - 1)] =
  (0. 3 + 0.3941)²/(0.3)²/29 + (0.3941)²/28 = 55.6939

Now that we know v’ we check the t-table and find t_{0.05(1),55.6939} = 1.672677 ⇒ N₀ rejected

Question 54

What increases statistical power?

Answer 54

These elements increase statistical power:

1) greater difference between µ and µ₀
2) larger α
3) larger n
4) smaller σ²
5) one-tailed tests

Question 55

For a one-tailed Mann-Whitney / Wilcoxon test, you have to decide which is the tail of interest. How does this work?

Answer 55

Question 56

What are the assumptions of one-sample t-tests?

Answer 56

1) Data are a random sample
2) Each data point is independent from each other
3) Data come from a normally-distributed population

Question 57

Note: One-sample-t tests are robust against non-normality as long as data are symmetrical.

Question 58

How are the statistical hypotheses written for testing the proability of getting different means from two populations?

Answer 58

H₀: µ₁ = µ₂
HA: µ₁ ≠ µ₂

OR

H₀: µ_{1 -} µ₂ = 0
HA: µ₁ - µ₂ ≠ 0

Question 59

What are the assumptions for a two-sample t-test?

Answer 59

1) data are random and independent
2) Both samples come from normally-distributed populations
3) Both populations have equal variances

Question 60

★ two-sample two-tailed t-test example

We want to test for a difference in protein concentration between two pea populations:

Results:
mean_fert = 24 g protein
SS_fert = 261 g²
n_fert = 30

mean<sub>unfert</sub> = 21.8 g protein
SS<sub>unfert</sub> = 320 g<sup>2</sup>
n<sub>unfert</sub> = 29

Answer 60

• H₀*: µ1 - µ2 = 0
• H_A*: µ1 - µ2 ≠ 0

s²_p = (SS_f + SS_u)/df_f +df_u = (261 + 320)/(29 + 28) = 10.193 g²

s_x̄f-x̄u = sqrt(s²_P/n_f + s²_p/n_u) =
sqrt(10.193/30+ 10.193/29) = 0.8314 g

t = (x̄_f - x̄_u)/s_{<em>x̄f - x̄u</em>} = (24 - 21.8)/0.8314 = 2.645

v = 57, t-critical = 2.0.

absolute value > critical value, so we reject the null hypothesis.

Question 61

For the following one-tailed test hypotheses, based on the relationship between the observed and critical t-values, when do you reject the null hypothesis?

1) H_A: µ₁ - µ₂ < 0
2) H_A: µ₁ - µ₂ > 0

Answer 61

1) H_A: µ₁ - µ₂ < 0
* H₀* is rejected if t ≤ t_α(1),v’

2) H_A: µ₁ - µ₂ > 0
* H₀* is rejected if t ≥ t_α(1),v’

Note that for a two-tailed test,
For a two-tailed test,
H_A: µ₁ - µ₂ ≠ 0
we reject H₀ if | t | ≥ t ≥ t_α(2),v’

Question 62

★ two-sample one-tailed t-test example

We want to test the hypothesis that bean protein concentration increases by at least 2 g/100 g beans when bean plants are fertilized. We do the study and get the following results:

mean_fert = 24 g protein
SS_fert = 261 g²
n_fert = 30
df_fert = 29

mean_unfert = 21.8 g protein
SS_unfert = 320 g²
n_unfert = 29
df_unfert = 28

Answer 62

• H₀*: µ_{<span>f</span>} - µ_{<span>u</span>}**H_A: µ_{<span>f</span>} - µ_{<span>u</span>} ≥ 2
• s²_p* = (SS_f + SS_u)/df_f +df_u = (261 + 320)/(29 + 28) = 10.193 g²

s_x̄f-x̄u = sqrt(s²_P/n_f + s²_p/n_u) =
sqrt(10.193/30 + 10.193/29) = 0.8314 g

t = (x̄_f - x̄_u)/s_x̄f-x̄u = (24 - 21.8 - 2)/0.8314 = 0.240558

v = 57, t-critical = 1.67.

Because out t-value is less than the t-critical, we cannot reject out null hypothesis that there is a difference of at least 2 g between both treatments.

Question 63

What assumption violations is the t-test most sensitive to?

Answer 63

T-test is quite robust to considerably non-normality, but violation of random/independence and homogeneity of variances is serious.

Question 64

8. For the figure below, in which two-sample t-test would there be higher power?

a) A
b) B

Answer 64

a) A
* *​b) B**

Question 65

9. Use the figure below to answer the next 3 questions. Which area under the curve(s) represents the probability of correctly not rejecting the null hypothesis?

A) A
B) B
C) C
D) D
E) A + D
F) C + B

Answer 65

A) A
B) B
C) C
D) D
E) A + D
​F) C + B

Question 66

10. In the figure above, which area under the curve(s) represents the probability of incorrectly not rejecting the null hypothesis?

A) A
B) B
C) C
D) D
E) A + D

Answer 66

A) A
B) B
C) C
D) D
E) A + D

Question 67

11. In the figure above, if this hypothesis test were performed at a significance level of 0.01, what probability would A represent?
    A) 0.05
    B) 0.975
    C) 0.01
    D) 0.0005
    E) 0.005

Answer 67

A) 0.05
B) 0.975
C) 0.01
D) 0.0005
​E) 0.005

Question 68

Which two factors increase rubustness against heterogenous variances in a t-test?

Answer 68

T-tests are a little bit more robust against variance heterogeneity if:

1) sample sizes are similar
2) sample sizes are above 30
2) the test is two-tailed

Question 69

How are assumptions of a two-sample t-test tested?

Answer 69

1) data are random and independent: cannot be checked. Done from experimental design.
2) Both samples come from normally-distributed populations: Visual inspection and Shapiro-wilk test

3) Both populations have equal variances:
Visual inspection and Fligner - Killeen test

Question 70

One example of violation of the independence assumption is when samples are paired (repeated measues). How could you get around the assumption of independence with paired data?

Answer 70

Paired data can be combined into a new sample by calulating their differences and this will now make data points independent.

Question 71

For two-sample analysis, how do you analyse the data in the following scenarios?

1) Both samples normal and equal variances
2) Both samples normal but unequal variances
3) Both samples non-normal but equal variances
4) Both sampes non-normal and unequal variances

Answer 71

1) Both samples normal and equal variances:
two-sample t-test with pooled variance

2) Both samples normal but unequal variances:
Welsch’s two-sample t-test (no pooled variance)

3) Both samples non-normal but equal variances:
Mann-Whitney or Wilcoxon rank test

4) Both sampes non-normal and unequal variances:
Transformation and re-assessment

Question 72

Note that for Welsch’s test, we use a t’ statistics instead of a t statistic.

Same as v’, which is a different calculation of df.

Question 73

What is the main characteristics of the Mann-Whitney / Wilcoxon test?

Answer 73

It’s a non-parametric test. Because of this:

1) It does not require estimation of population paameters
2) Hypotheses are not statements about population parameters

However,
3) it assumes that the data are random

Question 74

How are data treated in a Mann-Whitney / Wilcox test?

What is a drawback of this test?

Answer 74

Data are ranked wither from high to low or from low to high.

Convertion of data into ranks causes a loss of information and therefore power.

Question 75

For the following samples of germination times, fill in the “Rank A” and “Rank B” columns with the ranks that we would assign to these data in order to do a two-sample Mann-Whitney/Wilcoxon test.

Answer 75

Step 1: assign ranks to all numbers. If a number is repeated, they still get ranks n+1 where n is the previous rank.

Step 2: average the ranks in the repeated numbers.

Question 76

What are the two statistics calculated in a Mann-Whitney / Wilcoxon test?

How are they calculated?

Answer 76

u = n₁n₂ +[n₁(n₁ + 1)]/2 - R₁

u’ = n1n2 - u

Question 77

★Mann-Whitney / Wilcoxon test example

Height of males: 193, 188, 185, 183, 180, 175, 170
Height of females: 178, 173, 168, 156, 163
Ranks of male heighs: 1, 2, 3, 4, 5, 7, 9.
Ranks of female heighs: 6, 8 ,10, 11, 12

n<sub>m</sub> = 7
n<sub>f</sub> = 5
R<sub>m</sub> = 31
R<sub>f</sub> = 47

R is the sum of the ranks from each sample

Answer 77

• H₀* = Male and female students are the dame height
• H_A* = Male and female students are not the same height

Not that no hypothesis is made on any population parameters.

u = n₁n₂ + n₁(n₁ + 1)/2 - R₁
= (7)(5) + (7)(8)/2 - 31
= 35 + 28 - 31
= 32

u’ = n₁n₂ - u = (7)(5) - 32 = 3

Then you compare either u or u’, whichever is larger to the u critical (u_α(2),n1,n2). If greater, reject H₀.

This calculaton is not done by hand for in the exam.

Question 78

★Mann-Whitney / Wilcoxon test in R

How do you do this in R?

Answer 78

1) make a string will all the data:
height
2) make a string corresponding to sex for each data point
sex
3) test:
wilcox.test(height~sex)

Question 79

1. What is a t-value?
   a) A variance
   b) A number of standard errors from the mean for a t-distribution with a given number of degrees of freedom
   c) A statistic that, without any other information, tells you whether your alternative hypothesis is true
   d) A non-parametric test statistic

Answer 79

a) A variance

b) A number of standard errors from the mean for a t-distribution with a given number of degrees of freedom

c) A statistic that, without any other information, tells you whether your alternative hypothesis is true
d) A non-parametric test statistic

Question 80

2. On a standard normal distribution, 95% of the observations are contained within how many σ of μ? Choose the best approximation.

a) 1
b) 1.645
c) 2
d) 2.5
e) 3

Answer 80

a) 1
b) 1.645
c) 2
d) 2.5
​e) 3

Question 81

3. In which of the following situations should we select a Welch’s two-sample t-test as the most appropriate and powerful option for conducting a hypothesis test?

a) Both samples are non-normally distributed, sample variances are equal, and sample distributions are similar
b) One sample is non-normally distributed and variances are unequal
c) One sample is non-normally distributed and variances are not equal
d) Both samples are normally distributed, and variances are equal
e) Both samples are normally distributed and variances are unequal

Answer 81

a) Both samples are non-normally distributed, sample variances are equal, and sample distributions are similar
​b) One sample is non-normally distributed and variances are unequal
c) One sample is non-normally distributed and variances are not equal
d) Both samples are normally distributed, and variances are equal
e) Both samples are normally distributed and variances are unequal

Question 82

4. Which statement about the following study description is correct?

A herbicide-resistant strain of wheat and a non-herbicide resistant strain of wheat are grown, with 30 plants of each in a greenhouse before they are sprayed with a new herbicide that is going on the market. The researcher wants to test whether the herbicide-resistant strain (which was genetically engineered for resistance to different herbicides than the one being tested in this study) shows better growth and seed set than the control, following the spray.

a) The dependent variables are growth and seed set.
b) In a graph of the seed set results, seed set should be plotted on the x-axis.
c) A one-sample test is appropriate for this situation.
d) A paired-sample test is appropriate for this situation.

Answer 82

a) The dependent variables are growth and seed set.
b) In a graph of the seed set results, seed set should be plotted on the x-axis.
c) A one-sample test is appropriate for this situation.
​d) A paired-sample test is appropriate for this situation

Question 83

5. Which of the following statements is correct?

a) A statistical hypothesis is a statement about a cause-and-effect relationship between 2 or more variables.
b) A scientific hypothesis is a statement about a cause-and-effect relationship between 2 or more variables.
c) A statistical hypothesis must be proved to accept or reject a scientific hypothesis
d) “Descriptive statistics” refers to testing how much variation in an observed variable is due to a predictor variable, versus how much is due to chance alone.

Answer 83

a) A statistical hypothesis is a statement about a cause-and-effect relationship between 2 or more variables.
b) A scientific hypothesis is a statement about a cause-and-effect relationship between 2 or more variables.
c) A statistical hypothesis must be proved to accept or reject a scientific hypothesis
​d) “Descriptive statistics” refers to testing how much variation in an observed variable is due to a predictor variable, versus how much is due to chance alone.

Question 84

6. If we are interested in testing a hypothesis about a difference in two means, as the uncertainty (error) of our estimates of the means increases, our chance of detecting a real difference:

a) Decreases
b) Increases
c) Is not affected

Answer 84

• *a) Decreases**
  b) Increases
  c) Is not affected

Question 85

7. Conceptually, why is the standard error of the mean always smaller than the standard deviation of a sample, when both are derived from the same sample data?

a) Standard deviation is a measure of sample variability, whereas standard error of the mean is an estimate of the standard deviation of the distribution of sample means from which that sample is assumed to have come, and distributions of sample means are always narrower than the sample distribution from which they are estimated.
b) Standard deviation is not always smaller than the estimate of standard error derived from the same sample. It is bigger when sample size is large (>30).
c) Because the standard deviation represents the 95% confidence interval, whereas standard error represents one standard deviation of the distribution of sample means.
d) Standard deviation is the width of the distribution of sampling means, whereas standard error is a measure of sample variability, and the distribution of sample means is always more variable than a single sample.

Answer 85

a) Standard deviation is a measure of sample variability, whereas standard error of the mean is an estimate of the standard deviation of the distribution of sample means from which that sample is assumed to have come, and distributions of sample means are always narrower than the sample distribution from which they are estimated.
b) Standard deviation is not always smaller than the estimate of standard error derived from the same sample. It is bigger when sample size is large (>30).
c) Because the standard deviation represents the 95% confidence interval, whereas standard error represents one standard deviation of the distribution of sample means.
​d) Standard deviation is the width of the distribution of sampling means, whereas standard error is a measure of sample variability, and the distribution of sample means is always more variable than a single sample.

Question 86

1. What is the Central Limit Theorem?

Answer 86

States the following:
The distribution of means taken from a population which is or not normal will approximate normality as sample size increases.

Question 87

2. What is a p-value?

Answer 87

The probability of collecting certain data if H₀ was true.

Question 88

3. For a two-tailed t-test, with α of 0.05, what does the lower critical value tell us?

Answer 88

It tells us the t-value below which there is a 2.5% or lower chance of having gotten a sample t-value that small if the null hypothesis was true.

Question 89

The probability of rejecting a true H₀ is called Type ______ Error.

Answer 89

Type I error

Question 90

The probability of failing to reject a false H₀ is called Type _____ Error.

Answer 90

Type II error

Question 91

The Greek symbol for the proability of rejecting a true H₀ is

Answer 91

alpha

Question 92

The Greek symbol for the probability of failing to reject a false H₀ is

Answer 92

betta

Question 93

Power =

Answer 93

Power = 1 - betta

Question 94

List the 3 assumptions of a two-sample t-test.

Answer 94

1) data is random and independent
2) both samples are normally-distributed
3) both samples have equal variances

Question 95

note: there is no simple mathematical relationship beteen type I and type II error

Question 96

How are statistical hypotheses for a paired samples test written?

Answer 96

H_0: µ_d = µ0
H_A: µ_d ≠ µ0

where µ_d is the mean difference between pairs

Question 97

A test-statistic for two samples can be expressed as:

How does this scale up to mulptiple samples?

Answer 97

Question 98

Why does a test with multiple samples have to use two tails?

Answer 98

one-tail tests do not make sense when we have more than two samples.

Question 99

We conduct a study in which we raise a cohort of 36 goldfish in one large tank for one year. We then place 12 of the goldfish in a small pond, 12 in a medium-sized pond, and 12 in a large pond. We leave them in these ponds for 1 year, and then collect them all and measure their lengths.

What are we trying to determine using a test-staistic?

Answer 99

We use a test-statistic to see if the differences in the means that we found are statistically significant (i.e. if we would observe similar differences if the experiment was repeated.

Question 100

Based on the following image, how do we measure effect?

Answer 100

To measure effect we measure the distance of each group mean from the overall mean of all samples.

Differences (X̄_i - X̄),
where i is the group identifier and X̄ is the overall mean:
X̄₁ = - 1. 69
X̄₂ = 0.22
X̄₃ = 1.47

Question 101

Research hypothesis: adult goldfish grow larger when they live in larger ponds.

How do you write this as a statistical hypotesis if we have 3 samples?

Answer 101

• H₀*: Mean fish size is the same in all pond sizes
• H_A*: Mean fish size is not the same in all pond sizes

An often seen but incorrect way to write this:

• H₀*: μ_L = μ_M = μ_S
• H_A*: μ_L ≠ μ_M ≠ μ_S (not correct)

Question 102

Note that SS_among is the numerator of the t-statistic.

test-statistic = ss_among/error

Now we need the error for the denominator.

Question 103

Based on the following image, what is n, N, and k?

Answer 103

• n* = number of observatons (j = 1 to ni) within each group, n_i
• N* = total number of observations
• K* = number of groups (in this case ponds)
• n₁* = 12; n₂ = 12; n₃ = 12
• N* = 36
• k* = 3

Question 104

Based on the following image, what are the two kinds of error we can think about?

Answer 104

1) deviation of each observation from its group mean (SS_within)
2) deviation of each observation from the overall mean (SS_total)
1) Xi - X̄i
2) Xi - X̄,

where Xi is each observation, X̄i is group mean, and X̄ is overall mean

Question 105

The formula for SS_within is:

What is the formula for SS_total?

Answer 105

The formula for SS_total is:

Question 106

Note that SS_total equals the sum of SS_among and SS_within

SS_total = SS_among + SS_within

Question 107

Differences (X̄i - X̄),
X̄1 = - 1. 69
X̄2 = 0.22
X̄3 = 1.47

How do we avoid differences cancelling out?

Answer 107

to avoid differences calcelling each other pout, we use squares AND multiply by sample size to weigh;
Squared differences n_i(X̄i - X̄)²
X̄1 = 34.45
X̄2 = 0.59
X̄3 = 26.01

Then, summign that, we get:

SS_{among groups} = ∑n_i(X̄i - X̄)² = 61.06

Question 108

What is error in statistics?

Answer 108

Error is any deviation of an observation from the true mean of its population.

Question 109

How is the F-ratio aka F-statistic composed?

Answer 109

The F-statistic is composed of:
variance due to deviation of group means from overall mean (Effect), divided by variance due to deviation of each observation from its group mean (Error)

Note we’re dividing variances.

Question 110

The F-statistic has a known distribution.
What does the shape of the F-distribution depend on?

Answer 110

The shape of the F-distribution depends on the DF of the numerator and the DF of the denominator.

Question 111

What are the degrees fo freedom for
SS_total, SS_among, SS_within?

Answer 111

SS<sub>total</sub> = *N* -1
SS<sub>among</sub> = *k* - 1
SS<sub>within</sub> = *N* - *k*

Note that also DF_total = DF_among + DF_within

Question 112

Results from ANOVA are reported in ANOVA tables.
How would an ANOVA table look the previous example?
SSAmong groups = 61.06
DFAmong = 2
MSAmong = 30.53

SSWithin groups = 119.5
DFWithin = 33
MSWithin = 3.62

F = 8.43

Answer 112

Question 113

Instead of using SS, we use MS, which makes SS into variance terms.

How do you calculate MS?

Answer 113

MS_among = SS_among/DF_among

Question 114

We saw the following visual representation of the fish ANOVA data:

What is another way of plotting?

Answer 114

Another way of plotting is response variable on y-axis and predictor variable on x-axis.

Question 115

How do you calculate MS_error?

What are other names for MS_error?

Answer 115

MS_error = SS_within/DF_within
= SS_error/DF_error

MS_error is also called MS_within interchangeably.
MS_error also called residual.

Question 116

Why do we use DF as a denominator instead of sample size?

Answer 116

DF represents the number of observations available to estimate a parameter.

Question 117

What statistic do we use for a test with multiple samples?

Answer 117

F-statistic aka F-ratio

Question 118

Why is F-statistic also known as F-ratio?

Answer 118

The F-statistic is a ratio between two variances.

Question 119

Just like for the normal distribution, we can calculate the area under the curve for a given point or critical value.

What is the formula for F-critical?

Answer 119

F_crit = F_{α(1)DFamong,DF2within}

Question 120

What is a very important assumption of an F-statistic?

Answer 120

F assumes that the variances come from normally-distributed populations.

Question 121

For the fish size and lake size example:

SSAmong groups = 61.06
DFAmong = 2
MSAmong = 30.53

SSWithin groups = 119.5
DFWithin = 33
MSWithin = 3.62

Answer 121

F = MS_among/MS_within

F = 30.53/3.62 = 8.43
F<sub>0.01(1),2,33</sub> = 3.28 - (for some reason we used α of 0.01)

We reject the null hypothesis and conclude fish size is not equal across ponds.

Question 122

In a case where k = 2:

1) we could use either an F-test or a T-test and get the same result
2) MS_error = s²_p

3) F-value will equal the t-value squared
F_α(1)1,(N-2) = (t_α(2),(N-2))²

4) If a one-tailed test is required, the t-test is applicable, but ANOVA is not.

Question 123

Note that we use one tail in the notation of the F-value formula. This is because the F-distribution is assymetrical and has only one tail.

Question 124

Why do we need to do multiple comparisons?

Answer 124

To know which means are signifficantly different from one another.

Question 125

Why is it invalid to use multiple t-tests after an ANOVA?

Answer 125

Multiple t-tests inflate type I error.

Question 126

plotting the results of a Tukey test in R:
plot(TukeyHSD(model))

gives us 95% confidence intervals

Question 127

Results from Tukey tests are often plotted:

Question 128

Tukey test results also plotted with lines

Question 129

What test do you use if sample sizes are not equal?

How is it different from Tukey?

Answer 129

Tukey-Kramer test

Different from regulat Tukey beacuse it uses a different SE term:

Question 130

What happens if you want to do a Tukey or Tukey-Kramer test but the variances across samples are unequal?

Answer 130

Tukey is sensitive to different variances, so you can use the Welsch approximation for the Tukey test:

Question 131

If we wanted to test:

H₀: μ₁=μ₂ AND H₀: μ₂=μ₃ AND H₀: μ₁=μ₃

What is the proability of incorrectly rejecting at least one of the three H₀’s?

What is the problem with this?

Answer 131

the probability of incorrectly rejecting at least one of the H0’s is:

1− (1 − α)^C = 1 − (1 − 0.05)³ = 0.14,

where C is the number of possible different pairwise combinations of k samples

The problem is that 0.14 is much larger than 0.05

Question 132

What do multiple comparison procedures control for?

Answer 132

Multiple comparisons control for the experimentwise type I error by keeping it at α.

Question 133

What is the meaning of α when doing multiple comparisons?

Answer 133

for multiple comparisons, α is the probability of commiting at least one type I error

Question 134

What are the two options for peforming multiple comparisons?

Answer 134

1) posthoc comparisons
2) a priori (pre-planned) contrasts

Question 135

1) Specifically, what are posthoc comparisons used for?
2) Specifically, what are pre-planned constrasts used for?

Answer 135

1) posthoc comparisons are used to compare all pairs of means
2) pre-planned constrasts are used to rest a limited subset of hypotheses

Question 136

Tukey test aka honestly significantly different test (HSD) or wholly significant different (WSD).

Question 137

★ How is the MS of a contrast calculated?

Answer 137

float numbers are means of treatments
14 is n in every treatment

Question 138

Can Tukey test be used without doing an ANOVA?

Answer 138

Yes, Tukey tests can be performed without first doing an ANOVA.

Note that not all posthoc tests can.

Question 139

What is a disadvantage of doing a Tukey test after an ANOVA, instead of doing the Tukey test first?

Answer 139

Doing an ANOVA test before a Tukey test can lower statistical power.

Nonetheless, the common practice is to do the ANOVA and hen Tukey.

Question 140

What are the steps for doing a Tukey test?

Answer 140

Assuming that the two sample sizes are equal:

1) Arrange and number all sample means in order of increasing magnitude

2) Calculate pairwise differences between the means X̄_i – X̄_A
(Note _i is the group with the highest mean)

3) Calculate q-statistic: divide a difference between two means:
q = (X̄_B - X̄_A)/SE
where SE = sqrt(s²/n)
Note that you calculate a q for each comparison

4) H₀: X̄_B = X̄_A is rejected if q is greater than q-critical, q_α,df,k

Question 141

The conclusions of the Tukey test depend on the order in which the pairs of means are compared.

What is the proper procedure for comparing pairs of means?

Answer 141

1) Largest mean compared against smallest mean, then against second smallest, so on…
2) Second largest mean compared against smallest, then second smallest, so on…

Question 142

How is it demonstrated that the SS of contrasts is partitioned among the 3 orthogonal contrasts?

Answer 142

The SSamong equals the SS of the 3 contrasts added together:

Question 143

What can we conclude if there is no significant differnce between 2 means is found?

Answer 143

if no significant differnce between 2 means is found we can conclude that there are no significant differences between eclosed means

Question 144

Calculate of q-statistics for the fish experiment example:

Mean 1 = 3.917
Mean 2 = 5.833
Mean 3 = 7.983

SE = 0.549

q-crit = q_0.05,33,3 = 3.407

Answer 144

q-crit = q_0.05,33,3 = 3.407

• 3 vs 1:** (7.083 - 3.917)/0.549 = 3.166/0.549 = 5.767 ⇒ reject *H₀
• *3 vs 2:** (7.083 - 5.833)/0.549 = 1.25/0.549 = 2.277 ⇒ H₀ not rejected
• 2 vs 1:** (5.833 - 3.917)/0.549 = 1.916/0.549 = 3.49 ⇒ reject *H₀

Question 145

What is a disadvantage of using a priori tests instead of post hoc?

Answer 145

a priori tests do not allow comparisons of all pairs of means

Question 146

What is something a priori contrasts allow to do but post hoc don’t?

Answer 146

• a priori* contrasts allow to compare one mean against an average of other means
• a priori* constrasts are also more powerful than post-hoc

Question 147

Researchers are interesrted in determining whether there are positive effects of two common sponge species on the root growth of a mangrove tree.

Treatments:
A: unmanipulated
B: fake spongo
C: sponge species 1
D: sponge species 2

the ANOVA yielded a p-value of 0.003

Question 148

1) what does orthogonality mean?

Answer 148

1) orthogonality means that the contrasts are independent from one another

Question 149

example of orthogonal contrasts:

If living sponge tissue enhances mangrove root growth, then the average growth of the two living sponge treatments should be greater than the growth of roots in the inert foam treatment

How are the coefficients of these contrasts computed?

Answer 149

control (0)
fake sponge (2)
sponge spp 1 (-1)
sponge spp 2 (-1)

Question 150

what are the degrees of freedom of a contrast?

Answer 150

1 df

Question 151

★ how is the F-ratio of a priori contrasts calculated?

Answer 151

MS/MS_error =
0.145/0.164 = 0.882

Question 152

The formula for SSwithin is:

Question 153

why is orthogonality important?

Answer 153

ortogonality ensures that P-values are not inflated

Question 154

what are the rules for orthogonality?

Answer 154

1) sums of coefficients must equal 0
2) for k treatment groups, only k - 1 contrasts
3) the sum of cross-wise coefficients must also be 0

Question 155

In this example, are contrasts orthogonal?
Why?

contrast one:
control (0)
fake sponge (2)
sponge 1 (-1)
sponge 2 (-1)

contrast two:
control (3)
fake sponge (-1)
sponge 1 (-1)
sponge 2 (-1)

Answer 155

They are orthognal because:
1) their coefficient sums are 0 in both cases

2) their number is lower than k - 1

3) their cross-products equal 0:
(0) (3)+(2)(-1)+(-1)(-1)+(-1)(-1) = 0

Question 156

what are the components of ANOVA table?

Answer 156

1) source of variation
2) degrees fo freedom
3) sums of squares
4) mean squares
5) F
6) p- value

Question 157

What is a required conditionto use a priori contrasts?

Answer 157

they have to be determined before doing the statistical analysis

Question 158

what does sample non-normality suggest?

Answer 158

sample non-normalit suggests population non-normality

Question 159

What happens if the ANOVA assumptions are not met and an ANOVA is performed anyway?

Answer 159

the result of the ANOVA cannot be trusted if the assumptions are not met

Question 160

why are assumptions important for ANOVA?

what are the assumptons of ANOVA?

Answer 160

ANOVA is a parametric test

assumptions:

1) independent, random samples
2) all samples come from normal populations
3) variances between all treatments are equal

Question 161

What test is performed instead of ANOVA if sample variances are unequal but distributions are normal?

Answer 161

Welch’s ANOVA for unequal variances

Question 162

what test is performed instead of ANOVA if sample variances are equal but distributions are non-normal?

Answer 162

Kruskal-Wallis

Question 163

what is done if samples for an ANOVA are neither normal nor have equal variances?

Answer 163

transformation an assumption re-assessment

Question 164

how is the assumption of variance homogeneity checked?

Answer 164

1) Visual assessment (histograms or QQ-plots)
2) Fligner-Killeen test

Question 165

When can QQ-plots not be used?

Answer 165

QQ-plots are not adviced for samples with fewer than 25 observations. In that case, histograms are better

Question 166

how is the asumption of normality checked?

Answer 166

normality is checked through

1) assessment (hisrograms)
2) Shapito-Wilk test

Question 167

How does Kruskal-Wallis ranked test work?

Answer 167

1) assign ranks to observations
2) tied observations get average of the ranks they would get if not tied

Question 168

What is a disadvantage of the Kruskal-Wallis ranked test?

Answer 168

observations lose information when they are converted to ranks

Question 169

what is P-hacking?

Answer 169

trying different analyses until one is significant

Question 170

Statistically, why is P-hacking wrong?

Answer 170

P-hacking changes the actual α value

Question 171

Other ways fo P-hacking:

Answer 171

• taking too many data points
• not adjusting p-values for multiple comparisons

Question 174

What does alpha represent in a Tukey test?

Answer 174

In a Tukey test, alpha represents the probabiltiy of commiting at least one Type I error among all comparisons

Question 175

Note that:
when k=2, either an ANOVA or a t-test can be used, and the F value will equal the t vaue squared

However:
If a one-tailed test is required, then an ANOVA cannot be used

Question 176

how are data treated in linear regression?

Answer 176

pairs of data. x-values paired with y-values

Question 177

in a real life situation (where variation is present), what is the equation for linear regression?

Answer 177

Ŷi = α + βXi + εi

Question 178

how are data from linear regression plotted?

Answer 178

data from linear regression are plotted in a scatterplot.

Question 179

what is the equation for linear regression assuming a perfect model?

Answer 179

Ŷi = α + βXi

Question 180

in
Ŷi = α + βXi

1) what is α?
2) what is β?

Answer 180

1) α is the intercept (i.e. of Ŷi where the line crosses y axis
2) β is the slope of the line (i.e. the increase in Ŷi every unit of X)

Question 181

α and β are population parameters
How are they estimated?

Answer 181

we stimate α and β from our sample as a and b

Question 182

in Ŷi = α + βXi + εi,
what is ε

Answer 182

ε is error (i.e. the departure of an Yi from a Ŷi).

where Ŷi is what the equation predicts Yi to be

Question 183

What is the sum of all εi?

Answer 183

the sum of all εi equals 0

Question 184

what method is used to calculate the linear regression parameter estimates?

Answer 184

Least Squares

Question 185

what does Xi Yi mean?

Answer 185

Xi Yi is a singple point (pair of X and Y)

Question 186

what is Xi,Ŷi?

Answer 186

Xi,Ŷ is a point corresponding to an X that falls on the line of best fit

Question 187

what is the difference between Xi,Yi and Xi,Ŷi called?

Answer 187

the difference between Xi,Yi and Xi,Ŷi is called a residual

Question 188

what is the equation to calculate the slope?

Answer 188

Question 189

What is the equation to calculate the intercept?

Answer 189

we derive from the first equation to calculate α

Question 190

What happens if you change the inercept but not the slope?

Answer 190

Line moves up or down

note that a negative intercept makes the line cross X blow Y = 0

Question 191

what happens if you change the slope?

Answer 191

The line moves, but anchored at the intercept

Question 192

what does the Least Squares method calculate?

Answer 192

the Least Squares metod calcualtes the equation for the line that minimized differences between Y an Ŷ

Question 193

visually estimating the slope

Answer 193

Slope = (Y2 – Y1)/(X2 – X1)
Slope = (10 − 16) / (5 − 2)
Slope = (−6) / (3)
Slope = −2

Question 194

why should extrapolation not be done in regression?

Answer 194

The function does not hold infinitely, not awat from the intercept and not into the intercept either

Question 195

calculating α and β from two points

Answer 195

1) calculate β
2) use Ŷi = α + βXi

b = (20 – 0) / (3 – 1) = 10

Use the point (3,20) to calculate a:

a = Y – bX =
20 – 10*3 = -10

Question 196

Interpolation in linear regression is not wrong

Question 197

what population parameter are we interested in from linear regression?

Answer 197

we’re interested in β because that’s the parameter that defines the relationship between predictor and response variable

Question 198

1) determine variability of the response variable, SSY or SS_total
2) determine the amount of variability among Yi, “regression of sums of squares” or SSR or SSreg

Note: the last formulae are easiest for hand calculations

Answer 198

To obtain SSR we need:

Question 199

What is the difference between simple linear regression and simple linear correlation?

Answer 199

simple linear regression assumes dependence of oen variable upon another

in simple linear correlation there’s a relationship but not dependence

Question 200

What is residual error in a regression and how is it obtained?

Answer 200

residual error as a measure of the scatter of data points around the regression line.

Question 201

Using SSR, SSY, and SS_resid, we have partitioned the total variation in Yi into variation explained by the regression line and variation not explained by the regression line

SSY = SSR + SSE
AKA
SS_total = SS_regression + SS_residual

Answer 201

Note: the lines don’t add up because they are not yet squared

Question 202

What does β tell us?

Answer 202

β tells how much the response variable Y increases per unit increase of predictor variable X

Question 203

what are the assumptions for linear regression?

Answer 203

1) for each value of X, Y must be random an independent of one another
2) for each X, there exists a normal distribution of Y (and a normal distribution of ε
3) homogeneity of variances in the population (the variances of the distributions of Y values must all be equal)
4) relationship between X an Y is linear (mean of Yi lies in a straight line)
5) measurments of X are obtained without error (impossible, so we assume error is irrelevant)

Question 204

We use b, but what we;re really interested in is β

What is β?

Answer 204

β is the functional dependence in the population

Question 205

what are the hypotheses for linear regression?

Answer 205

H0: β = 0
HA: β ≠ 0

Question 206

How is the value of r² interpreted?

Answer 206

r² = 1 means all the variation in Y is explained by X

r² = 0 means all the variation in Y is explained by X

Question 207

How can hypotheses about β be tested?

Answer 207

ANOVA or t-test method

Note: testing anything other than H0: β =0 (e.g. H0: β - β0) requries that we use a t-test

Question 208

Regression using t-test for hypothesis about a:

Question 209

SSR will be equal to SSY only if each data point falls on the regression line (very unlikely).

Question 210

How are the DF calcualted in regression?

Answer 210

DF<sub>reg</sub> = 1
DF<sub>total</sub> = n - 1
DF<sub>resid</sub> = n - 2

Question 211

with the DF you can now calculate the MS’s

MS<sub>reg</sub> = SS<sub>reg</sub>/DF<sub>reg </sub>
MS<sub>resid</sub> = SS<sub>resid</sub>/DF<sub>resid</sub>
F = MS<sub>reg</sub> / MS<sub>resid</sub>

Question 212

What is r², aka coefficient of determination?

Answer 212

r² indicates how strong the relationship is
aka
how much of the total variation in Y is attributed to X

Question 213

How is r² calculated?

Answer 213

r² = SSreg/SStotal = SSR/SSY

Question 214

Regression using t-test for hypothesis about b:
t = (b - β0) / S_b

where S_b is
Sb = sqrt(s²_XY * SSX)

OR
Sb = sqrt(MS_resid/SSX)
and
MSresid = sum(Yi - Ȳi)²/(n -2)

Question 215

Formulae for t-critical

t_α(2),n-2
t_α(1),n-2

Question 216

What is the concept of a degree of freedom?

Answer 216

if you calculate the mean from a set of numbers n, one of thos numbers is not free to variable and df is n - 1

Question 217

what isone definition of degrees of freedom?

Answer 217

the number of values in the final calculation of a statistic that are free to vary int he daata sample

the maximum number of logicaly independent values

Question 218

What are the two types of degrees of freedom?

Answer 218

DF associated with the effect of interest
DF associated with the error

Question 219

1) What are DF in ANOVA?
2) What are DF in regression?

Answer 219

ANOVA: DFgroups = k - 1 (where k is the number of groups)

Regression: DF_reg = 1

Question 220

In linear regression, why does DFreg = 1?

Answer 220

In regression we only calculate 1 parameter more than the mean of Y (remember the mean of Y is a).

Ȳ = a + bx

Question 221

what is the generalformula for error DF?

Answer 221

DFerror = n - p

Where n is sample size and p is the number of parameters used for estimations

For regression, DFerror = n-2 becaue we need to know a and b