Question

In: Statistics and Probability

10-65. Consider the hypothesis test H0: σ_1^2=σ_2^2 against H1: σ_1^2<σ_2^2. Suppose that the sample sizes are...

10-65. Consider the hypothesis test H0: σ_1^2=σ_2^2 against H1: σ_1^2<σ_2^2. Suppose that the sample sizes are n1 = 6 and n2 = 13, and that s_1^2=23.8 and s_2^2=27.8. Use α = 0.05. Test the hypothesis and explain how the test could be conducted with a confidence interval on σ_1^2/σ_2^2.

Solutions

Expert Solution

The provided sample variances are and and the sample sizes are given by and

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:

Ha:

This corresponds to a left-tailed test, for which a F-test for two population variances needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the the rejection region for this left-tailed test is R={F:F<FL​=0.214}.

(3) Test Statistics

The F-statistic is computed as follows:

(4) Decision about the null hypothesis

Since from the sample information we get that F=0.856≥FL​=0.214, it is then concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population variance ​ is less than the population variance ​, at the α=0.05 significance level.

Confidence Interval

The 95% confidence interval for ​​ is: 0.22<​<5.586.

Please let me know in comments in case anything is unclear. Will reply ASAP. Do upvote if satisfied!


Related Solutions

10-65. Consider the hypothesis test H0: σ_1^2=σ_2^2 against H1: σ_1^2<σ_2^2. Suppose that the sample sizes are...
10-65. Consider the hypothesis test H0: σ_1^2=σ_2^2 against H1: σ_1^2<σ_2^2. Suppose that the sample sizes are n1 = 6 and n2 = 13, and that s_1^2=23.8 and s_2^2=27.8. Use α = 0.05. Test the hypothesis and explain how the test could be conducted with a confidence interval on σ_1^2/σ_2^2.
Consider the following hypothesis test. H0:  = 20 H1:  ≠ 20 The sample size...
Consider the following hypothesis test. H0:  = 20 H1:  ≠ 20 The sample size is 200 and the standard deviation of the population is 10. Use a  = 0.05. What is the probability of making the Type II error if the real value of the population is: a. μ = 18.0 b. μ = 22.5 c. μ = 21.0
Given the following hypothesis: H0 : μ ≤ 10 H1 : μ > 10 For a...
Given the following hypothesis: H0 : μ ≤ 10 H1 : μ > 10 For a random sample of 10 observations, the sample mean was 11 and the sample standard deviation 4.20. Using the .10 significance level: (a) State the decision rule. (Round your answer to 3 decimal places.) _____ H0 if t > (b) Compute the value of the test statistic. (Round your answer to 2 decimal places.) Value of the test statistic ____ (c) What is your decision...
Suppose H0: mu = 120 is tested against H1: mu > 120 with known variance 10...
Suppose H0: mu = 120 is tested against H1: mu > 120 with known variance 10 and n = 25. What p-value corresponds to a sample mean ybar = 121.5
Suppose that we are to conduct the following hypothesis test: H0:μ =1030 H1:μ>1030 Suppose that you...
Suppose that we are to conduct the following hypothesis test: H0:μ =1030 H1:μ>1030 Suppose that you also know that σ=170, n=90, x¯=1058.9, and take α=0.01. Draw the sampling distribution, and use it to determine each of the following: A. The value of the standardized test statistic: Note: For the next part, your answer should use interval notation. An answer of the form (−∞,a)(−∞,a) is expressed (-infty, a), an answer of the form (b,∞)(b,∞) is expressed (b, infty), and an answer...
Consider the following: H0: μ=10 H1: μ≠10 For a sample of 6 units, the mean for...
Consider the following: H0: μ=10 H1: μ≠10 For a sample of 6 units, the mean for the sample is 11.2 and the population standard deviation is 2.14. 1)Calculate the p-value for the test. 2) If α=0.10, do you reject the null hypothesis?
Consider the following hypothesis test: H0: ? = 16 Ha: ? ? 16 A sample of...
Consider the following hypothesis test: H0: ? = 16 Ha: ? ? 16 A sample of 40 provided a sample mean of 14.17. The population standard deviation is 6. a. Compute the value of the test statistic (to 2 decimals). b. What is the p-value (to 4 decimals)? c. Using ? = .05, can it be concluded that the population mean is not equal to 16? Answer the next three questions using the critical value approach. d. Using ? =...
Suppose we want to test the null hypothesis H0 : p = 0.28 against the alternative...
Suppose we want to test the null hypothesis H0 : p = 0.28 against the alternative hypothesis H1 : p ≠ 0.28. Suppose also that we observed 100 successes in a random sample of 400 subjects and the level of significance is 0.05. What are the critical values for this test? a. -1.96 and 1.96 b. 0.05 and 0.01 c. -1.39 and 1.39 d. -1.6449 and 1.6449
Suppose we wish to test the hypothesis H0 :μ=45vs. H1 :μ>45. What will be the result...
Suppose we wish to test the hypothesis H0 :μ=45vs. H1 :μ>45. What will be the result if we conclude that the mean is 45 when the actual mean is 50? Choose one of the following. 1. We have made a Type I error. 2. We have made a Type II error. 3. We have made the correct decision.
Suppose that we are to conduct the following hypothesis test: H0:μ=1010 H1:μ>1010 Suppose that you also...
Suppose that we are to conduct the following hypothesis test: H0:μ=1010 H1:μ>1010 Suppose that you also know that σ=250, n=95, x¯=1067.5, and take α=0.01. Draw the sampling distribution, and use it to determine each of the following: A. The value of the standardized test statistic: Note: For the next part, your answer should use interval notation. An answer of the form (−∞,a) is expressed (-infty, a), an answer of the form (b,∞) is expressed (b, infty), and an answer of...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT