Question 1

Suppose that a t test of H₀ : μ = 250 against H₁ : μ > 250 is based on 10 degrees of freedom.If the calculated value of the test statistic is t = 2.228,then the p-value for this test is 0.025.

Accepted Answer

Since the calculated t-value is positive and the alternative hypothesis is one-sided (μ > 250), we will find the p-value corresponding to the right-tail of the t-distribution with 10 degrees of freedom. Using a t-table or calculator, we can find that the area to the right of t=2.228 is approximately 0.025. Therefore, the p-value for this test is 0.025, which means we can reject the null hypothesis at the 0.05 level of significance.

Question 2

Suppose a test procedure about the population mean μ is performed,when the population is normal with an unknown variance,and a sample of size 20 is drawn from the population.If the alternative hypothesis is H₁ : μ > μ₀,then the null hypothesis is rejected at a level α = 0.05 if t > 1.729.

Accepted Answer

This is true because if the calculated t-value is greater than the critical value of 1.729, then we reject the null hypothesis and accept the alternative hypothesis. This is true for a one-tailed test with a significance level of 0.05.

Question 3

Explain the relationship between Type I error and Type II error in a hypothesis test.

Accepted Answer

The answer of Explain the relationship between Type I error...

Question 4

For any significance level α,the two-sided rejection region Z ≥ z_α/2or Z ≤ -z_α/2 has a Type I error probability equal to 1 - α.

Accepted Answer

The Type I error probability for a two-sided test at significance level α is exactly α, not 1 - α. The 1 - α represents the confidence level, not the Type I error probability.

Question 5

Differentiate between a one-sided and two-sided composite alternative hypothesis when the null hypothesis is H₀ : μ = 10?

Accepted Answer

The answer of Differentiate between a one-sided and two-sided composite...

Question 6

Give an account of how a decision is made concerning the null hypothesis.

Accepted Answer

The answer of Give an account of how a decision...

Question 7

Suppose that a test procedure about the population proportion P is performed,and that the sample proportion is approximately normally distributed.If the alternative hypothesis is H₁ : P < P₀,then the rejection region for a level α is Z < -z_α.

Accepted Answer

This statement correctly describes the rejection region for a one-tailed test with a lower alternative hypothesis. The rejection region is in the left tail of the standard normal distribution, and the cutoff value is determined by the selected significance level (α) and the corresponding z-score from the standard normal distribution (denoted as -zα). Therefore, if the test statistic (calculated from the sample proportion) falls in this left tail region (i.e., the z-score is less than -zα), we reject the null hypothesis in favor of the alternative hypothesis.

Question 8

Suppose a test procedure about the population mean μ is performed,when the population is normal with an unknown variance.If a sample of size 16 is drawn from the population and the alternative hypothesis is H₁ : μ ≠ μ₀,the null hypothesis is rejected at a level α = 0.05 if either t < -2.131 or t > 2.131.

Accepted Answer

This is true based on the information given. The test procedure is for the population mean μ, the population is assumed to be normal with an unknown variance, a sample of size 16 is drawn, and the alternative hypothesis is a two-sided alternative (μ can be less than or greater than the hypothesized value). The critical values for a two-tailed test with α = 0.05 and degrees of freedom = 15 are -2.131 and 2.131, which are provided in the statement of the problem. Therefore, if the calculated t-value falls outside of this range, the null hypothesis is rejected at the 0.05 level.

Question 9

If the p-value of a two-sided test for the mean of a population is .005,then the null hypothesis will be rejected at a level of significance of .01.

Accepted Answer

The answer of If the p-value of a two-sided test...

Question 10

Explain the distinction between Type I and Type II errors.

Accepted Answer

The answer of Explain the distinction between Type I and...

Question 11

The supervisor of a production line believes that the average time to assemble an electronic component is 14 minutes.Assume that assembly time is normally distributed with a standard deviation of 3.4 minutes.The supervisor times the assembly of 14 components,and finds that the average time for completion was 11.6 minutes.Is there evidence that the average amount of time required to assemble a component is something other than 14 minutes? Use α = 0.01.

Accepted Answer

The answer of The supervisor of a production line believes...

Question 12

An alternate hypothesis is a statement about a population parameter that is accepted only when the null hypothesis is rejected.

Accepted Answer

This is because when we reject the null hypothesis, it means that the alternative hypothesis is more likely to be true. Therefore, the acceptance of the alternative hypothesis is dependent on the rejection of the null hypothesis.

Question 13

Explain carefully the distinction between simple and composite hypotheses.

Accepted Answer

The answer of Explain carefully the distinction between simple and...

Question 14

If a level of significance is less than or equal to the p-value,the null hypothesis is not rejected.

Accepted Answer

This statement is true. The level of significance represents the maximum probability of rejecting the null hypothesis when it is actually true. The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming that the null hypothesis is true. If the level of significance is less than or equal to the p-value, it means that the observed test statistic is unlikely to have occurred by chance alone, and the null hypothesis is not rejected.

Question 15

A music industry professional claims that the average amount of money that an average teenager spends per month on music is at least $50.Based upon previous research,the population standard deviation is estimated to be $12.42.The music professional surveys 20 students and finds that the mean spending is $47.77.Is there evidence that the average amount spent by students is less than $50?

Accepted Answer

The answer of A music industry professional claims that the...

Question 16

If the null hypothesis is P ≥ 0.20 and the alternate hypothesis is P < 0.20,then a two-tail test is being conducted.

Accepted Answer

A two-tail test would have an alternate hypothesis that is not specific to either direction (e.g. P ≠ 0.20). In this case, the alternate hypothesis specifically states that the true proportion is less than 0.20, indicating a one-tail test.

Question 17

If a random sample of size n is drawn from a normal distribution,then the standardized
variable t = (  
- μ)/ (s/  
)has a Student's t-distribution with n-1 degrees of freedom.

Accepted Answer

This is correct according to the definition of the Student's t-distribution. When a random sample is drawn from a normal distribution, the t-statistic is used to test hypotheses about the mean of the population. Its distribution follows the Student's t-distribution with n-1 degrees of freedom.

Question 18

Suppose that a test procedure about the population proportion P is performed,and that the sample proportion is approximately normally distributed.If the alternative hypothesis is H₁ : P ≠ P₀,then the rejection region for a level α = 0.05 is either Z < -1.96 or Z > 1.96.

Accepted Answer

The rejection region for a two-tailed test with a level of significance α = 0.05 is Z < -1.96 or Z > 1.96. This means that if the calculated Z-score falls outside of this range, the null hypothesis can be rejected in favor of the alternative hypothesis.

Question 19

When testing the mean of a normal population,the Z test is always used.

Accepted Answer

While the Z test can be used when the population standard deviation is known, if the population standard deviation is unknown and the sample size is small (typically less than 30), then the t-test should be used instead.

Question 20

If the calculated value of the test statistic for a two-sided test is Z = 1.84,then the p-value is 0.0329.

Accepted Answer

The p-value for a two-sided test with Z = 1.84 is approximately 0.0662, not 0.0329. This is because the p-value should be calculated for both tails of the normal distribution, and 0.0329 represents the area in one tail beyond Z = 1.84.

Quiz 9: Hypothesis Testing: Single Population

Suppose that a t test of H0 : μ = 250 against H1 : μ > 250 is based on 10 degrees of freedom.If the calculated value of the test statistic is t = 2.228,then the p-value for this test is 0.025.

Suppose a test procedure about the population mean μ is performed,when the population is normal with an unknown variance,and a sample of size 20 is drawn from the population.If the alternative hypothesis is H1 : μ > μ0,then the null hypothesis is rejected at a level α = 0.05 if t > 1.729.