Module 4: Tests for Hypothesis (Theory)

A) H0 : μ = 50,000; H0​ : μ = 50,000; Reject H0​ if z ≥ 1.645

A researcher claims that the average salary of entry-level data analysts is $50,000. A random sample of 35 data analysts has a mean salary of $51,500 with a known population standard deviation of $4,800. Test this claim at a 0.05 significance level. Which of the following correctly represents the null hypothesis and the decision criteria?

A) H0 : μ = 50,000; H0​ : μ = 50,000; Reject H0​ if z ≥ 1.645
B) H0 : μ ≤ 50,000; H0​ : μ ≤ 50,000; Reject H0 if z ≤ −1.645
C) H0 : μ ≥ 50,000; H0​ : μ ≥ 50,000; Reject H0​ if z ≥ 1.96
D) H0 : μ = 50,000; H0​ : μ = 50,000; Reject H0​ if z ≥ 1.96

A) ±1.96

Given a two-tailed test with a significance level of 0.05, what critical z-value should be used?

A) ±1.96
B) ±1.645
C) ±2.33
D) ±1.28

B) t-test for the mean

In a hypothesis test regarding the mean of a population where the sample size is 25 and the population variance is unknown, which of the following tests should you use?

A) z-test for the mean
B) t-test for the mean
C) χ² test for variance
D) F-test for two variances

B) H0 : σ2 = 225; χ² = 21.2

A company claims its product lifetime follows a normal distribution with a variance of 225 hours. A sample of 16 products showed a variance of 265 hours. At a 0.05 significance level, which of the following correctly states the null hypothesis and the test statistic?

A) H0 : σ2 = 225; χ² = 18.7
B) H0 : σ2 = 225; χ² = 21.2
C) H0 : σ2 = 225; χ² = 15.3
D) H0 : σ2 ≠ 225; χ² = 22.5

C) z-test; critical value = -2.33

A restaurant manager believes the average cost of a meal is less than $30. A sample of 40 meals had an average cost of $29 with a standard deviation of $3. Which type of test and critical value should the manager use for a left-tailed test at a 0.01 significance level?

A) z-test; critical value = -1.645
B) t-test; critical value = -2.704
C) z-test; critical value = -2.33
D) t-test; critical value = -2.576

D) Increasing the sample size can help reduce the probability of Type II error.

Which of the following statements is true regarding Type I and Type II errors?

A) Type I error occurs when we fail to reject the null hypothesis when it is true.
B) Type II error occurs when we reject the null hypothesis when it is false.
C) The probability of committing a Type I error is denoted by β.
D) Increasing the sample size can help reduce the probability of Type II error.

C) This is a right-tailed test.

For a hypothesis test, which of the following is correct if the null hypothesis states H0 : μ ≤ 100 and the alternative hypothesis is H1 : μ > 100?

A) This is a two-tailed test.
B) This is a left-tailed test.
C) This is a right-tailed test.
D) None of the above.

C) χ² test, df = 24

A cigarette manufacturer claims the variance in nicotine content of its product is 0.5 mg. A sample of 25 cigarettes showed a variance of 0.8 mg. At the 0.05 level, which test and degree of freedom should be used?

A) t-test, df = 24
B) z-test, df = 25
C) χ² test, df = 24
D) F-test, df = 23

B) A conjecture that no difference exists or that a parameter equals a specific value

Which of the following best describes the null hypothesis (H0)?

A) A statement that suggests a significant difference exists between groups

B) A conjecture that no difference exists or that a parameter equals a specific value

C) The hypothesis tested only with a z-test

D) A hypothesis that assumes a mean difference

A) There is a 1% chance of rejecting the null hypothesis

If a hypothesis test is conducted at the 0.01 significance level, what does this mean?

A) There is a 1% chance of rejecting the null hypothesis

B) There id a 1% of rejecting the null hypothesis when it is false

C) The probability of committing a type II error is 0.01

D) The null hypothesis will definitely be rejected if p = 0.01

C) Type I error

What type of error occurs when the null hypothesis is rejected even though it is true?

A) Type II error
B) Beta error
C) Type I error
D) Power error

B) t-test

Which of the following tests is appropriate when the population standard deviation is unknown, and the sample size is less than 30?

A) z-test
B) t-test
C) Chi-squared test
D) F-test

C) The parameter is either greater than or less than a specific value.

A one-tailed test is used to test a hypothesis that:

A) The parameter is equal to a specific value.
B) There is no difference between groups.
C) The parameter is either greater than or less than a specific value.
D) The parameter equals the population mean.

B) Reject H0

if the p-value is less than the significance level in a hypothesis test, what decision is made?

A) Accept H0H
B) Reject H0
C) Accept H1
D) Fail to reject H1

B) A known population standard deviation

A sample mean is compared to a known population mean using a z-test. What is a primary requirement for this test?

A) A sample size smaller than 30
B) A known population standard deviation
C) A significance level of 0.01
D) A two-tailed test

C) F-test

Which test is appropriate for comparing the variances of two samples?

A) z-test
B) t-test
C) F-test
D) Chi-squared test

B) -1.645

A researcher is conducting a left-tailed test at the 0.05 significance level. What is the critical z-value?

A) -1.96
B) -1.645
C) 1.96
D) 1.645

A) ±1.28

In a two-tailed test at the 0.10 significance level, what is the critical value for z?

A) ±1.28
B) ±1.645
C) ±1.96
D) ±2.33

C) It approaches a normal distribution as sample size increases.

Which of the following statements is true about the t-distribution?

A) It is used when the population variance is known.
B) It has a mean of 0 and variance of 1 regardless of sample size.
C) It approaches a normal distribution as sample size increases.
D) It is always skewed to the right

B) The data should follow a normal distribution.

If a test for variance uses the chi-squared distribution, what must be true?

A) The sample must have at least 30 observations.
B) The data should follow a normal distribution.
C) The test is always one-tailed.
D) The sample mean is known.

C) Two-sample t-testm

Which hypothesis test is used to compare the means of two independent samples with unknown variances?

A) z-test
B) Paired t-test
C) Two-sample t-test
D) F-test

C) Failing to reject H0H0​ when it is false

In a hypothesis test, which of the following scenarios would be a Type II error?

A) Rejecting H0H0​ when it is false
B) Failing to reject H0H0​ when it is true
C) Failing to reject H0H0​ when it is false
D) Accepting H1H1​ when it is false

A) If the variance equals a specific value

What does a chi-squared test for variance determine?

A) If the variance equals a specific value
B) If two sample variances are equal
C) If the mean of a sample differs from a known population mean
D) If the distribution is normal

C) When testing if a parameter is greater than a specified value

When should a right-tailed test be used?

A) When testing if a parameter is equal to a specified value
B) When testing if a parameter is less than a specified value
C) When testing if a parameter is greater than a specified value
D) For all non-directional hypotheses

C) The probability of rejecting H0H0​ when it is false

What does the power of a test refer to?

A) The likelihood of rejecting H0H0​ when it is true
B) The probability of failing to reject H0H0​ when it is false
C) The probability of rejecting H0H0​ when it is false
D) The level of significance of the test

C) Values for which H0​ is rejected

The critical region in a hypothesis test represents:

A) Values for which H0​ is not rejected
B) The probability of a Type II error
C) Values for which H0​ is rejected
D) The significance level of the test

D) μ ≠ k

In a two-tailed test, the alternative hypothesis is expressed as:

A) μ > k
B) μ < k
C) μ = k
D) μ ≠ k

B) It will decrease

If a sample size increases while all other factors remain constant, what is likely to happen to the p-value?

A) It will increase
B) It will decrease
C) It will stay the same
D) It will reach 0