a. To find a 98% confidence interval for the mean μ of a normal population with variance 9 from the sample 32, 42, 41, 35, 48, and 55, we can use the formula: Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size) First, let's calculate the sample mean: Sample mean = (32 + 42 + 41 + 35 + 48 + 55) / 6 = 253 / 6 = 42.17 Next, we need to find the critical value. Since we want a 98% confidence interval, we need to find the z-score corresponding to a 99% confidence level (since the z-score corresponds to the area outside the confidence interval). Using a z-table or calculator, we find the z-score to be approximately 2.33. Now, we need to calculate the standard deviation: Standard deviation = square root of variance = square root of 9 = 3 Finally, we can calculate the confidence interval: Confidence interval = 42.17 ± (2.33 * 3 / square root of 6) b. To find a 98% confidence interval for the mean μ of a normal population from the sample 32, 42, 41, 35, 48, and 55, we need to know the sample variance. Without the sample variance, we cannot calculate the confidence interval. c. To test the hypothesis σ2 = σ02 = 10 against three kinds of alternatives (σ2 σ02, σ2 ≠ σ02) using a sample size n = 15 and a sample variance s2 = 13 from a normal population, we can perform an F-test. The F-test compares the ratio of the sample variance to the hypothesized population variance. The test statistic follows an F-distribution. First, let's calculate the test statistic: Test statistic = (sample variance / hypothesized population variance) * ((n - 1) / (n - k)) Where k is the number of parameters being estimated. In this case, k = 1 since we are estimating the population variance. Test statistic = (13 / 10) * ((15 - 1) / (15 - 1)) = 1.3 Next, we need to find the critical value from the F-distribution table or calculator. Since we are testing at a significance level of 1%, we need to find the critical value that corresponds to a 99% confidence level. Let's assume the critical value is F_critical. Finally, we compare the test statistic to the critical value: If test statistic F_critical, we reject the null hypothesis. If test statistic

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a. To find a 98% confidence interval for the mean μ of a normal population with variance 9 from the sample 32, 42, 41, 35, 48, and 55, we can use the formula: Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size) First, let's calculate the sample mean: Sample mean = (32 + 42 + 41 + 35 + 48 + 55) / 6 = 253 / 6 = 42.17 Next, we need to find the critical value. Since we want a 98% confidence interval, we need to find the z-score corresponding to a 99% confidence level (since the z-score corresponds to the area outside the confidence interval). Using a z-table or calculator, we find the z-score to be approximately 2.33. Now, we need to calculate the standard deviation: Standard deviation = square root of variance = square root of 9 = 3 Finally, we can calculate the confidence interval: Confidence interval = 42.17 ± (2.33 * 3 / square root of 6) b. To find a 98% confidence interval for the mean μ of a normal population from the sample 32, 42, 41, 35, 48, and 55, we need to know the sample variance. Without the sample variance, we cannot calculate the confidence interval. c. To test the hypothesis σ2 = σ02 = 10 against three kinds of alternatives (σ2 < σ02, σ2 > σ02, σ2 ≠ σ02) using a sample size n = 15 and a sample variance s2 = 13 from a normal population, we can perform an F-test. The F-test compares the ratio of the sample variance to the hypothesized population variance. The test statistic follows an F-distribution. First, let's calculate the test statistic: Test statistic = (sample variance / hypothesized population variance) * ((n - 1) / (n - k)) Where k is the number of parameters being estimated. In this case, k = 1 since we are estimating the population variance. Test statistic = (13 / 10) * ((15 - 1) / (15 - 1)) = 1.3 Next, we need to find the critical value from the F-distribution table or calculator. Since we are testing at a significance level of 1%, we need to find the critical value that corresponds to a 99% confidence level. Let's assume the critical value is F_critical. Finally, we compare the test statistic to the critical value: If test statistic > F_critical, we reject the null hypothesis. If test statistic < F_critical, we fail to reject the null hypothesis. Note: The specific values for the test statistic and critical value will depend on the degrees of freedom, which are calculated as (n - 1) for the numerator and (n - k) for the denominator.

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a. To find a 98% confidence interval for the mean μ of a normal population with variance 9 from the sample 32, 42, 41, 35, 48, and 55, we can use the formula: Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size) First, let's calculate the sample mean: Sample mean = (32 + 42 + 41 + 35 + 48 + 55) / 6 = 253 / 6 = 42.17 Next, we need to find the critical value. Since we want a 98% confidence interval, we need to find the z-score corresponding to a 99% confidence level (since the z-score corresponds to the area outside the confidence interval). Using a z-table or calculator, we find the z-score to be approximately 2.33. Now, we need to calculate the standard deviation: Standard deviation = square root of variance = square root of 9 = 3 Finally, we can calculate the confidence interval: Confidence interval = 42.17 ± (2.33 * 3 / square root of 6) b. To find a 98% confidence interval for the mean μ of a normal population from the sample 32, 42, 41, 35, 48, and 55, we need to know the sample variance. Without the sample variance, we cannot calculate the confidence interval. c. To test the hypothesis σ2 = σ02 = 10 against three kinds of alternatives (σ2 < σ02, σ2 > σ02, σ2 ≠ σ02) using a sample size n = 15 and a sample variance s2 = 13 from a normal population, we can perform an F-test. The F-test compares the ratio of the sample variance to the hypothesized population variance. The test statistic follows an F-distribution. First, let's calculate the test statistic: Test statistic = (sample variance / hypothesized population variance) * ((n - 1) / (n - k)) Where k is the number of parameters being estimated. In this case, k = 1 since we are estimating the population variance. Test statistic = (13 / 10) * ((15 - 1) / (15 - 1)) = 1.3 Next, we need to find the critical value from the F-distribution table or calculator. Since we are testing at a significance level of 1%, we need to find the critical value that corresponds to a 99% confidence level. Let's assume the critical value is F_critical. Finally, we compare the test statistic to the critical value: If test statistic > F_critical, we reject the null hypothesis. If test statistic < F_critical, we fail to reject the null hypothesis. Note: The specific values for the test statistic and critical value will depend on the degrees of freedom, which are calculated as (n - 1) for the numerator and (n - k) for the denominator.

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