MODULE 8.1: hypothesis testing

Question 1

For a hypothesis test with a probability of a Type II error of 60% and a probability of a Type I error of 5%, interpret the meaning

Answer 1

There is a 5% probability that the null hypothesis will be rejected when actually true, and the probability of rejecting the null when it is false is 40%.

Question 2

What is a hypothesis in the context of hypothesis testing?

Answer 2

A hypothesis is a statement about the value of a population parameter developed for testing a theory or belief, typically stated in terms of parameters like the population mean (μ).

Question 3

Q: What is the purpose of hypothesis testing procedures?

Answer 3

A: To determine whether a hypothesis is reasonable and should not be rejected or if it is unreasonable and should be rejected.

Question 4

Q: What is the null hypothesis (H0)?

Answer 4

A: The null hypothesis is the hypothesis that the researcher wants to reject, typically stated as a simple statement about a population parameter, such as
H0 = u = u0
.

Question 5

Q: What condition does the null hypothesis always include?

Answer 5

= condition

Question 6

Q: What is the alternative hypothesis (Ha)?

Answer 6

A: The alternative hypothesis is what is concluded if there is sufficient evidence to reject the null hypothesis. It is mutually exclusive and exhaustive with the null hypothesis.

Question 7

Q: What are the critical z-values for a two-tailed test at a 5% significance level (α = 0.05)?

Answer 7

A: The critical z-values are ±1.96.

Question 8

Q: When do we reject the null hypothesis in a two-tailed z-test?

Answer 8

A: If the computed test statistic falls outside the range of critical z-values (test statistic > 1.96 or < -1.96).

Question 9

Q: What are Type I and Type II errors in hypothesis testing?

Answer 9

Type I error: Rejecting the null hypothesis when it is true.
Type II error: Failing to reject the null hypothesis when it is false.

Question 10

Q: What does the significance level (α) represent?

Answer 10

A: The significance level is the probability of making a Type I error (rejecting a true null hypothesis).

Question 11

Q: What is the power of a test?

Answer 11

The probability of correctly rejecting the null hypothesis when it is false, calculated as 1 - P(type II error)

Question 12

Q: How is the test statistic calculated?

Answer 12

A: The test statistic is the difference between the sample statistic and the hypothesized value, scaled by the standard error of the sample statistic.

Question 13

Q: What distributions can a test statistic follow?

Answer 13

A: The t-distribution, z-distribution (standard normal), chi-square distribution, and F-distribution.

Question 14

Q: How does sample size affect the probability of Type II error?

Answer 14

A: Increasing the sample size decreases the probability of a Type II error, thus increasing the power of the test.

Question 15

Q: Why is it incorrect to say “accept” the null hypothesis?

Answer 15

A: Because the null hypothesis can only be supported or rejected, not proven.

Question 16

Q: What must be determined before calculating the critical value for a hypothesis test?

Answer 16

A: Whether the test is one-tailed or two-tailed, the significance level, and the distribution of the test statistic.

Question 17

Q: What is the standard error of the sample statistic when the population standard deviation (σ) is known?

Answer 17

SE = SD / SQRT(n)

Question 18

Q: What does a two-tailed hypothesis test evaluate?

Answer 18

A: Whether the sample statistic is significantly different from the hypothesized value in either direction (greater or less).

Question 19

Q: What does the decision rule for a two-tailed z-test at α = 0.05 state?

Answer 19

A: Reject the null hypothesis if the test statistic > 1.96 or < -1.96; otherwise, fail to reject the null.

Question 20

Q: What is the relationship between the significance level (α) and the probability of a Type II error?

Answer 20

Decreasing α (e.g., from 0.05 to 0.01) increases the probability of a Type II error.

Question 21

Q: What are the characteristics of the null and alternative hypotheses?

Answer 21

A: They are mutually exclusive (no overlap) and exhaustive (cover all possible outcomes).

Question 22

Q: Why is the critical value important in hypothesis testing?

Answer 22

A: It is the threshold against which the computed test statistic is compared to decide whether to reject the null hypothesis.

Question 23

Q: What does the computed test statistic represent?

Answer 23

A: It represents how far the sample statistic is from the hypothesized value in standard error units.

Question 24

Q: What happens to the power of a test when the sample size increases?

Answer 24

A: The power of the test increases because the probability of a Type II error decreases.

Question 25

Q: What is the meaning of a 5% level of significance (α = 0.05)?

Answer 25

A: It means there is a 5% probability of rejecting a true null hypothesis (Type I error).

Question 26

Q: What does it mean for a test statistic to follow a z-distribution?

Answer 26

A: The test statistic is based on a standard normal distribution with a mean of 0 and a standard deviation of 1.

Question 27

Q: How is the critical value for a test statistic determined?

Answer 27

A: It is based on the chosen significance level (α) and the distribution of the test statistic (e.g., z-distribution, t-distribution).

Question 28

Q: What is the role of the cumulative probability table in hypothesis testing?

Answer 28

It provides critical z-values or probabilities for standard normal distribution, used to determine rejection regions.

Question 29

Q: What is the implication of failing to reject the null hypothesis?

Answer 29

A: The sample statistic is not sufficiently different from the hypothesized value, so the null hypothesis is supported.

Question 30

Q: Why must the type of hypothesis test (one-tailed or two-tailed) be determined in advance?

Answer 30

A: It affects the critical values and rejection regions used in the decision rule.

Question 31

Q: What is the primary goal of hypothesis testing?

Answer 31

A: To make inferences about population parameters based on sample data.

Question 32

Q: What happens to the probability of a Type II error when the significance level is increased?

Answer 32

A: The probability of a Type II error decreases, increasing the power of the test.

significance level = type 1 error. increasing significance means more probability of rejecting the null incorrectly

rejection region enlarges so really easy to reject null

Question 33

Q: What is the formula for the power of a test?

Answer 33

power = 1 - P(type II error)

Question 34

describe the central limit theorem

Answer 34

if u = 1%, then the distribution of the sample means is a t-distribution with n-1 degrees of freedom, mean = u, and dispersion equal to the standard error,
SE = x / SQRT(n)

Question 35

What does the significance level really mean (for example, a significance of 95%)

Answer 35

It means that 95% of the time, if our mean is the true mean, my sample means should fall within about 2 standard errors (or critical value) away from the hypothesized mean

Question 36

How do we know if a test is two tailed or one tailed?

Answer 36

you have to look at the null hypothesis. Does it have a >= or <= condition? If so, it’s one tailed .

If the null says u = something, then it’s two tailed