classic stat tests

Question 1

Hypothesis test for comparing the variances of two samples

Answer 1

f-test

Question 2

## F = 28.517, num df = 5, denom df = 4, p-value = 0.003169
## alternative hypothesis: true ratio of variances is greater than 1
## 95 percent confidence interval:
## 4.558378 Inf
## sample estimates:
## ratio of variances
## 28.51747

How can we interpret the R output?

Answer 2

It returns the following:

the value of the F test statistic.
the degrees of freedom of the F distribution of the test statistic.
the p-value of the test is 0.2105
95% confidence interval for the ratio of the population variances.
the ratio of the sample variances is 2.4081
The p-value of the F-test is p = 0.003169 which is lower than the alpha level of 0.05. In conclusion, there is a difference between the two samples.

Question 3

hypothesis test for a single mean

Answer 3

t-test

Question 4

One Sample t-test
## data: type_A
## t = -0.64336, df = 5, p-value = 0.5483
## alternative hypothesis: true mean is not equal to 70
## 95 percent confidence interval:
## 28.37016 94.96317
## sample estimates:
## mean of x
## 61.66667

How can we interpret the R output?

Answer 4

p-value: The two-tailed p-value that corresponds to a t test-statistic of -0.64336 and 5 degrees of freedom.

The null and alternative hypotheses for this one sample t-test are as follows:

H0: µ = 70 (the mean is 70)

HA: µ ≠ 70 (the mean is not 70)

Because the p-value of our test (0.5483) is greater than 0.05, we fail to reject the null hypothesis of the test.

Question 5

how to compare two means of the two groups.

Answer 5

two sample t-test

Question 6

what assumptions are made in two sample t-test

Answer 6

1. The two groups are independent;
2. The observations within each group are independent;
3. The variance of the measurements within each group are similar;
4. The observations from each of the groups are normally distributed.

Question 7

Welch Two Sample t-test
##
## data: type_A and type_B
## t = -0.82687, df = 5.4177, p-value = 0.4432
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -44.14995 22.28328
## sample estimates:
## mean of x mean of y
## 61.66667 72.60000**
Question How can we interpret the R output?

Answer 7

t: This is the t test-statistic.
The two hypotheses for this particular two sample t-test are as follows:

H0: µ1 = µ2 (the two population means are equal)
HA: µ1 ≠µ2 (the two population means are not equal)

Because the p-value of our test (0.4432) is more than alpha = 0.05, we accept the null hypothesis of the test.

Question 8

The Mann-Whitney U test (Wilcoxon rank-sum test)

Answer 8

test is used as a substitution for the unpaired t-test, when the assumptions of normality fail to be satisfied.

Question 9

what does the Mann-Whitney U test (Wilcoxon rank-sum test) compare

Answer 9

the median values in the two groups and not the mean like the t-test.

Question 10

what assumption is made in the Mann-Whitney U test (Wilcoxon rank-sum test)

Answer 10

the data can be ranked. Instead of using the observations, we use their ranking.
However, if the data is normally distributed using the Mann-Whitney test would be less powerful than using the t-test for small samples.

Question 11

Wilcoxon rank sum exact test
##
## data: type_A and type_B
## W = 13, p-value = 0.7922
## alternative hypothesis: true location shift is not equal to 0
Question How can we interpret the R output?

Answer 11

p-value of 0.7922 and the significance value we chose is 0.05, then there
is no evidence to reject the null hypothesis.

Question 12

comparing two dependent means

Answer 12

paired t-test

Question 13

Paired t-test
##
## data: group1 and group2
## t = 1.633, df = 4, p-value = 0.0889
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## -0.6109684 Inf
## sample estimates:
## mean of the differences
## 2

1. How do we interpret the R output?

Answer 13

p value is greater than 0.05; accept the null hypothesis

Question 14

what assumptions are made in paried t-test

Answer 14

1. Independence: Each observation should be independent of every other observation.
2. Normality: The differences between the pairs should be approximately normally distributed.
3. No Extreme Outliers: There should be no extreme outliers in the differences.

Question 15

what test is applied to matched or dependent samples

Answer 15

The Wilcoxon matched pairs test (Wilcoxon signed rank
test)

Question 16

Wilcoxon signed rank test with continuity correction
##
## data: group_1 and group_2
## V = 15, p-value = 0.05791
## alternative hypothesis: true location shift is not equal to 0

How do we interpret the R output?

Answer 16

p value is greater than 0.05; accept the null hypothesis

Question 17

A binomial test

Answer 17

when an experiment has two possible outcomes (i.e. success/failure) and you have an idea about what the probability of success is. A binomial test is run to see if observed test results differ from what was expected.

Question 18

##
## data: 90 and 200
## number of successes = 90, number of trials = 200, p-value = 0.179
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
## 0.3797536 0.5217507
## sample estimates:
## probability of success
## 0.45

Answer 18

The p-value of the test is 0.179. Since this is greater than 0.05, we can accept the null hypothesis

Question 19

Assumptions for the Binomial Test

Answer 19

Items are dichotomous (i.e. there are two of them) and nominal.
The sample size is significantly less than the population size.
The sample is a fair representation of the population.
Sample items are independent(one item has no bearing on the probability of another).

Question 20

one sample z-test is used to

Answer 20

test whether the mean of a population is less than, greater than, or equal to some specific value.

Question 21

what does the z-test assume

Answer 21

assumes that the population standard deviation is known.
The data are continuous (not discrete).
The data is a simple random sample from the population of interest.
The data in the population is approximately normally distributed.

Question 22

two sample z-test is used to

Answer 22

test whether two population means are equal.

Question 23

what assumptions are made in two sample z-test

Answer 23

The data from each population are continuous (not discrete).
Each sample is a simple random sample from the population of interest.
The data in each population is approximately normally distributed.
The population standard deviations are known.

Question 24

comparing counts in contigency tables

Answer 24

Pearson’s chi-squared test

Question 25

the hypothesis of a Pearson’s chi-squared test

Answer 25

H0: (null hypothesis) The two variables are independent.
H1: (alternative hypothesis) The two variables are not independent.

Question 26

##
## data: count
## X-squared = 35.334, df = 1, p-value = 2.778e-09

How can we interpret the R output?

Answer 26

p value less than 0.05 so refect null hypothesis

Question 27

Fisher’s Exact test

Answer 27

is to test if there is association between variables in a contigency table concept, but for small samples

Question 28

The power of a test is affected by a number of different things
(4)

Answer 28

1. The significance level. Tests with a lower level have a lower power.
2. The sample size. A larger sample size means more power.
3. The standard deviation of the data. If the data are very spread, it is harder to tell the difference between two means - hence the power is lower.
4. The effect size. i.e. the true difference between the two groups. The larger this is, the easier it is to
   spot in the data, and hence, the larger the power.

Question 29

What are the differences and similiraties between the unpaired t-test and the Mann-Whitney U test?

Answer 29

The Mann-Whitney test is used as a substitution for the unpaired t-test, when the assumptions of normality fail to be satisfied.

compares the median values in the two groups and not the mean like the t-test.

Question 30

Is the Wilcoxon signed rank test, a parametric test?

Answer 30

no, its a Non-parametric test
same as The Mann-Whitney U test (Wilcoxon rank-sum test)

Question 31

What are the Type I and Type II errors?

Answer 31

type I is faslse positive
type II is false negative

Question 32

nonparametric test doesn’t assume

Answer 32

normality

Question 33

what kind of test is used ff the data is not normally distributed but are symmetrically distributed around
the mean

Answer 33

valid to use parametric tests.

This is because the central limit theorem,
the means of samples which are symmetrically distributed tend to be normally distributed

if the sample size is large enough

Question 34

what is done if the data is postive or negative skewed

Answer 34

transformation to
make the distribution more symmetrical

Question 35

25 students took a test; the mean was predicited to be 58%
what test would used to test the different?

Answer 35

one sample t-test

Question 36

Students were tested on their ability to predict how moving bodies behave, both before
and after attending a course on Newtonian physics.
what test would see if attending the course have a significant effect on their test scores

Answer 36

paired t-test

Question 37

The pH of cactus cells was measured at dawn and at dusk using microprobes.
The cactus was identifiable, and two sets of measurements were carried out on it.
what test to anaylse?

Answer 37

two sample t-test

Question 38

We can never use a Normal approximation to test whether two proportions are equal or not.

Answer 38

false

Question 39

compare the proportion in one random sample to a specified population proportion

Answer 39

normal approximation