Exam 2 Study Guide

Question 1

What is the definition of probability?

Answer 1

• The likelihood of a certain event occuring
• p=specified outcome/total outcomes

Question 3

What is the role of probability in inferential statistics?

Answer 3

• Probability is used to obtain the likelihood of obtaining a specific sample from a given population
• If the probability of getting a specific sample is small, we can say that the sample probably came from a different population

Question 4

What are the requirements of random sampling?

Answer 4

Requires that each individual in the population has an equal chance of being selected

Question 5

What is the difference between sampling with replacement and sampling without replacement?

Answer 5

• Sampling with replacement requires each individual to be returned back to the population before making the next selection to keep probability constant
• Sampling without replacement does not require each individual to be returned back to the population, does not have requirement of constant probability

Question 6

Percentile vs Percentile rank

Answer 6

Percentile is a score, percentile rank is a percentage

Question 7

What makes a distribution of sample means normal?

Answer 7

• If the population the sample means are obtained from is normal
• If n is 30 or greater

Question 8

sampling error

Answer 8

the natural discrepency (amount of error) between a population parameter and the corresponding sample statistic

Question 9

The expected value of M

Answer 9

The mean of the DOSM is equal to the population mean (mu)

Question 10

Distribution of sample means

Answer 10

Distribution of Sample Means - the set of sample means for all possible random samples of specific size (n) that can be selected from a
population.
* This distribution has well-defined and predictable characteristics that are specified in the Central
Limit Theorem

Question 11

Sampling distribution

Answer 11

A distribution of statistics obtained by selecting all the possible samples of a specific size from a population

Question 12

Sampling distributions characteristics

Answer 12

• Sample means should pile up around the population mean; most sample means should be relatively close to the population mean
• Pile of samples should tend to form a normal shaped distribution (frequencies should taper off as the distance between M and mu increases)
• The larger the sample size, the closer the sample means is to mu (more representative). Thus, the sample means obtained with a large sample size should cluster relatively close to the population mean; the means obtained from small samples should be more widely scattered

Question 13

Central Limit Theorem (Shape, Central Tendency, Variability)

Answer 13

1. Mean of the theoretical distribution of sample means is called the Expected value of M (always equal to the population mean)
2. The standard deviation of the theoretical distribution of sample means is called the Standard error of M and is computed by oM (0/Square root of n)
3. The shape of the distribution of sample means is typically normal
4. Distribution of sample means approaches a normal distribution as n approaches infinity

Question 14

Law of large numbers

Answer 14

The larger the sample size (n) in a specific sample, the more probable that M is close to mu (larger sample, smaller standard deviation)

Question 15

Standard Error of the Mean (how to calculate, and what it measures)

Answer 15

• Provides a measurement of the average expected distance between the M and u (like standard deviation of DOSM)
• Describes the distribution of sample means (variability)
  1. Describes the distribution of sample means; provides a measure of how much difference is expected from one sample to another
  2. Measures how well an individual sample mean represents the entire distribution

Formula: Om= O / n squared

Decreases when sample size increases

Question 16

What is the critical value of z for a two tailed significance test w a=.05?

Answer 16

+ or - 1.96

Question 17

What are the goals of hypothesis testing?

Answer 17

• to use sample
  data to evaluate a hypothesis about a population
• to rule out chance (sampling error) as a plausible explanation for the results from a research study (although it does not actually do this)

Question 18

Null hypothesis (How to state using symbols and words)

for two-tailed tests

Answer 18

• When using two-tailed test, the null hypothesis symbol is that we predict M=mu
• means the observed findings are due to random chance (there does not appear to be a real effect)
• Ex. H0: Thinking Cap is not related to IQ. (μ = 100)

Question 19

Alternative hypotheses (how to state w symbols and words)

for two-tailed tests

Answer 19

• Symbol: mu=/ to M
• Means the observed findings cannot be explained by sampling error (there does appear to be a real effect).
  – Ex. H1: Thinking Cap is related to IQ. (μ ≠ 100).

Question 20

Process of hypothesis testing (4-step method)

Answer 20

1. State your hypothesis about the population
2. Define the probability at which we think the results indicate that there must be a true effect (Critical value: Defined by associations that are very unlikely to obtain(typically less than 5% chance) if no effect exists.)
3. Obtain and test a sample from the population
4. Compare data w the hypothesis predictions and make a decision to reject or fail to reject null

Question 21

Test statistic

Answer 21

• (ex. z-score) forms a ratio comparing the difference between the M and μ versus the amount of difference we would expect without any treatment
  effect (σM).

Question 22

What factors influence a hypothesis test?

Answer 22

The size of the treatment effect and the size of the sample… even a very small effect can be statistically significant if observed in a very large sample (would have a very small standard error)

Question 23

a (alpha) level

Answer 23

establishes a criterion or cut off for deciding if the null hypothesis is correct (typically a=.05)
* outcome of a hypothesis test can be influenced by alpha level because the smaller the alpha level, the less likely it is that values fall in the critical region (so you are less likely to reject the null hypothesis)

Question 24

Type 1 Error

Answer 24

• Occurs when sample data indicate an effect when no effect actually exists
• Rejecting the null hypothesis when the null is true.
• Caused by unusual, unrepresentative samples, falling in the critical region without any true effect.
• Hypothesis tests are structured to make Type I errors unlikely.

Question 25

Type II Error

Answer 25

• Occur when the hypothesis test
  does not indicate an effect but in reality an effect does exist.
  – We fail to reject the null hypothesis even though it was actually false.
  – More likely with a small treatment effect or poor study design (sample size too small).

Question 26

Directional/One-tailed tests critical region

Answer 26

• Critical region is 10% of scores but all on one side
• z=+ or - 1.645

Question 27

One-tailed test hypotheses

Answer 27

Example
* The original population has μ = 75 points on a stats test and studying is predicted to increase the scores:
-H0: Test scores are not increased.
-H1: Test scores are increased.
* Entire critical region would be located in the positive tail because large values for M would demonstrate that there is an increase and that we
would reject the null hypothesis

Question 28

How likely are you to commit a Type I error when p=.05?

Answer 28

20-50% depending on context

Question 29

What does a p-value measure?

Answer 29

the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis

do not tell us the probability of making a Type I Error

Question 30

How is a p-value used in hypothesis testing?

Answer 30

Imagine we’re testing a vaccine and our hypothesis test yields p = .05
* Incorrect: If you reject the null hypothesis, there’s a 5% chance that
you’re making a mistake.
– Actually, it’s probably about 20-50% depending on context.
* Correct: Assuming the vaccine had no effect, you’d obtain the observed
difference or more in 5% of studies due to random sampling error.

When performing hypothesis tests we will check statistical significance by seeing if our test scores (ex. z-score) indicate a p-value of less than our α level.

Question 31

Why do we calculate effect size in addition to statistical significance?

Answer 31

-Even a very small effect can be statistically significant if observed in a very large sample (would have a very small standard error)
* Effect size is a measure of the absolute magnitude of an effect, independent of sample size.

Question 32

How to calculate and interpret Cohen’s d?

Answer 32

• Like a z-test, Cohen’s d measures mean difference in terms of
  the standard deviation.
  -Small effect: d=.2-.49
  -Med effect: d=.5-.79
  -Large effect: d=.8+

Formula: Cohen’s d= (𝑀 − μ) / σ

Cohen’s d is a standardized effect size

Question 33

What is the power of a hypothesis test?

Answer 33

• the probability that the test will reject the null hypothesis when there is actually an effect (the power of avoiding a type II error)

Question 34

Why is the power of a hypothesis test important?

Answer 34

• It helps avoid a Type II error (is the probability of avoiding a Type II error)

Question 35

What factors influence statistical power?

Answer 35

Likelihood we can find what we are looking for depends on:
* effect size (larger effects are easier to find)
* sample size (larger samples make it easier to find effects)
* alpha level (larger alpha level makes it easier to find effects)
* non-directional vs directional hypothesis (directional tests make it
easier to find effects)

Question 36

When is a t-test used instead of a z-score?

Answer 36

• When we have no knowledge of the population standard deviation (but we do need to know/guess something about the population mean)
• Can be used for a completley unknown population, when our only available information comes from the sample

All that is required for a t-test is a sample and a reasonable hypothesis about the population mean

Question 37

Assumptions that should be true (or close to true) when using the t-statistic:

Answer 37

1. The data are continuous
2. The sample data have been randomly sampled from the population
3. The variability of the data in each group is similar
4. The sampled population is approximately normally distributed

Question 38

Where does the population mean come from in a one-sample t-test?

Answer 38

• Can be estimated from previous research or theory
• Can also be chosed to represent a defined and meaningful threshold (arbitrary)

Question 39

Estimated Standard Error (how to calculate and what it means)

Answer 39

• Estimated standard error is how much difference between the sample and population mean is reasonable to expect (like standard error of the mean, but o is unknown)
• Use standard deviation of sample instead of standard deviation of the population to calculate

Question 40

One sample t-test formula

Answer 40

• Forms a ratio: top contains the difference betweent the M and mu (aka the signal)
• Bottom is the estimated standard error (SM) (aka the noise)

Question 41

The t-statistic and the influence of sample size

Answer 41

• Think of t-value as “estimated z-test” bc we are using the sample sd to estimate the unknown population sd
• With large samples, the t-value will be very similar to a z-test
• With small samples, the t-value will provide a relatively poor estimate of z.

Question 42

Degrees of freedom and the t-distribution

Answer 42

• For the one-sample t test, degrees of freedom: df=n-1
• This describes how well the t statistic represents a z-test
• Also determines how well the t-distribution approximates a normal distribution
• For large values of df, the t distribution will be nearly normal, but with small values for df, it will be flatter and more spread out

Question 43

Hypothesis tests with the t-statistic

Answer 43

1. State the hypotheses and select a value for a (Note: the null always states a value for mu)
2. Locate the critical region (note: you must find the value for df and use to t distribution table)
3. Calculate the test statistic
4. Make a decision (reject or fail to reject the null hypothesis)

Question 44

Cohen’s d for t-statistic

Answer 44

Cohen′ s 𝑑= (𝑀 − μ) / 𝒔

effect levels are same boundaries

Question 45

What does r squared measure and how to calculate/interpret it?

Answer 45

• r^2 is the percentage of variance accounted for by the independent variable
• Measures the amount of variability that can be attributed to the IV, so we obtain a new measure of effect size
• Small effect: between 0.01-0.08
• Med effect: .09-.24
• Large effect: .25+

Question 46

Confidence intervals

Answer 46

A confidence interval is an interval, or range of values centered around a sample statistic. The logic behind a confidence interval is that a sample statistic, such as a sample mean, should be relatively near to the corresponding population parameter. Therefore, we can confidently estimate that the value of the parameter should be located in the interval near to the statistic.