Week 5

Question 1

Define “the distribution of sample means”

Answer 1

The collection of sample means for all the possible random samples of a certain size (n) that can be obtained from a population.

Question 2

Describe the logically predictable characteristics of the distribution of sample means.

Answer 2

1. Sample means should be close to population mean.
2. Sample means tend to form normally shaped distributions.
3. Larger sample sizes produce sample means closer to the population mean.

Question 3

Define “sampling distribution”

Answer 3

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

Question 4

What is the central limit theorem?

Answer 4

A mathematical theorem that specifies the characteristics of the distribution of sample means.

Question 5

Define “expected value of M”.

Answer 5

The mean of the distribution of sample means is equal to the mean of the population of scores. Denoted as μM.

Question 6

What does σM represent?

Answer 6

The standard error of M, which is the standard deviation of the distribution of sample means.
It shows ​the average distance between M and µ that would be expected if H0 was true.

Question 7

What is the law of large numbers?

Answer 7

As the sample size increases, the error between the sample mean and the population mean should decrease.

Question 8

What is the formula for σM?

Answer 8

σM=σ/square root of n.
OR
σM=square root of (σ²/n).

Question 9

How do you calculate the z-score for a sample mean?

Answer 9

z=M-μ/σM

rather than z=M-μ/σ

Question 10

Describe how the magnitude of the standard error is related to the size of the sample.

Answer 10

Small samples are normally associated with a large standard error, larger the sample, smaller the standard error.
If n=1 the standard error will match the population standard deviation.

Question 11

What does p̂ represent?

Answer 11

The sample proportion.

Question 12

What does p represent?

Answer 12

The population proportion.

Question 13

What is the sampling distribution of a proportion?

Answer 13

The distribution of sample proportions for all possible samples of a certain size which can be drawn from the population.

Question 14

What is the expected value of the mean of the sampling distribution of proportion?

Answer 14

The mean of the sampling distributions of proportion is equal to the population proportion.
(μp̂ = p)

Question 15

What is the equation used to find the standard error for the distribution of sample proportions.

Answer 15

𝜎p̂ = square root of (𝑝 𝘹 (1−𝑝)/n).

Question 16

What does 𝜎p̂ represent?

Answer 16

Standard error for the distribution of sample proportions.

Question 17

What equations must be true for the distribution of sample proportions to be normal? (shape)

Answer 17

𝑛𝑝 ≥ 10 AND 𝑛(1 − 𝑝) ≥ 10.

Question 18

How can the distribution of sample proportions be used for inference?

Answer 18

When the distribution of sample proportions is normally distributed, 95% of sample
proportions will lie within 1.96 standard errors of the population proportion.
This indicates what is considered a normal score, anything outside of this suggests it is from a different population.

Question 19

Define a “hypothesis test”.

Answer 19

A statistical method that uses sample data to evaluate a hypothesis about a population.

Question 20

What are the four steps of a hypothesis test?

Answer 20

1. State alternative (scientific) hypothesis (H₁) and null hypothesis (H₀).
2. Set the criteria for a decision (using alpha level).
3. Collect data and compute sample statistics (should only occur AFTER hypothesises are stated and criteria is set).
4. Make a decision.

Question 21

What does α represent?

Answer 21

The alpha level (level of significance) determines a probability value called the critical region, if sample data falls in the critical region (very unlikely outcomes), the null hypothesis is rejected.

Question 22

Define a Type I error.

Answer 22

Occurs when a researcher rejects a null hypothesis that is actually true (due to extreme samples).
The probability of a type I error is equal to the alpha level.

Question 23

Define a Type II error.

Answer 23

Failing to reject a false null hypothesis, which leads to failure to detect a real treatment effect.
The probability of a type II error is determined by Beta (statistical power).

Question 24

What should you take into consideration when selecting an alpha level?

Answer 24

1. Minimum should be 0.5.

2. Lower alpha levels will reduce type I errors but require more evidence.

Question 25

What is a directional/one-tailed hypothesis test?

Answer 25

1. The directional prediction is incorporated into the statement of the hypotheses.
2. The critical region is located entirely in one tail of the distribution.

Question 26

How do you calculate the probability that μ is equal to M?

Answer 26

Frequency of an M score divided by the total possible number of samples of a particular size.

Question 27

What is p-value?

Answer 27

Alternative to using z-score values to make hypothesis decisions.
Indicates how likely it is to get a result if the null hypothesis is true.
If the p-value is lower than the alpha level, you can reject H₀.