<p>a) To infer if the population mean is less than 100 at the 10% significance level, we can perform a one-sample t-test.</p> <p>Step 1: State the null hypothesis (H0) and the alternative hypothesis (H1). H0: μ = 100 (The population mean is equal to 100) H1: μ < 100 (The population mean is less than 100)</p> <p>Step 2: Calculate the test statistic. The formula for the t statistic is:</p> <p>t = (x̄ - μ) / (s/√n)</p> <p>where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.</p> <p>t = (75 - 100) / (50/√8) = -1.414</p> <p>Step 3: Determine the critical value from the t-distribution table. For a one-tailed test at the 10% significance level with 7 degrees of freedom (n-1), the critical value is -1.415.</p> <p>Step 4: Compare the test statistic with the critical value. If the test statistic is less than the critical value, we reject the null hypothesis. In this case, -1.414 is greater than -1.415, so we do not reject the null hypothesis. Therefore, we cannot infer at the 10% significance level that the population mean is less than 100.</p> <p>b) If we know that the population standard deviation is 50, we can perform a z-test instead of a t-test.</p> <p>Step 1: The null and alternative hypotheses are the same as in part (a).</p> <p>Step 2: Calculate the z statistic using the formula:</p> <p>z = (x̄ - μ) / (σ/√n)</p> <p>where σ is the population standard deviation.</p> <p>z = (75 - 100) / (50/√8) = -1.414</p> <p>Step 3: Determine the critical value from the z-distribution table. For a one-tailed test at the 10% significance level, the critical value is -1.28.</p> <p>Step 4: Compare the z statistic with the critical value. In this case, -1.414 is less than -1.28, so we reject the null hypothesis. Therefore, we can infer at the 10% significance level that the population mean is less than 100.</p> <p>c) The test statistics differ because a t-test is used when the population standard deviation is unknown and is estimated from the sample, while a z-test is used when the population standard deviation is known. The t-distribution is wider and has heavier tails than the z-distribution, which leads to a larger critical value and a higher chance of not rejecting the null hypothesis.</p> ####
Question
<p>a) To infer if the population mean is less than 100 at the 10% significance level, we can perform a one-sample t-test.</p> <p>Step 1: State the null hypothesis (H0) and the alternative hypothesis (H1). H0: μ = 100 (The population mean is equal to 100) H1: μ < 100 (The population mean is less than 100)</p> <p>Step 2: Calculate the test statistic. The formula for the t statistic is:</p> <p>t = (x̄ - μ) / (s/√n)</p> <p>where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.</p> <p>t = (75 - 100) / (50/√8) = -1.414</p> <p>Step 3: Determine the critical value from the t-distribution table. For a one-tailed test at the 10% significance level with 7 degrees of freedom (n-1), the critical value is -1.415.</p> <p>Step 4: Compare the test statistic with the critical value. If the test statistic is less than the critical value, we reject the null hypothesis. In this case, -1.414 is greater than -1.415, so we do not reject the null hypothesis. Therefore, we cannot infer at the 10% significance level that the population mean is less than 100.</p> <p>b) If we know that the population standard deviation is 50, we can perform a z-test instead of a t-test.</p> <p>Step 1: The null and alternative hypotheses are the same as in part (a).</p> <p>Step 2: Calculate the z statistic using the formula:</p> <p>z = (x̄ - μ) / (σ/√n)</p> <p>where σ is the population standard deviation.</p> <p>z = (75 - 100) / (50/√8) = -1.414</p> <p>Step 3: Determine the critical value from the z-distribution table. For a one-tailed test at the 10% significance level, the critical value is -1.28.</p> <p>Step 4: Compare the z statistic with the critical value. In this case, -1.414 is less than -1.28, so we reject the null hypothesis. Therefore, we can infer at the 10% significance level that the population mean is less than 100.</p> <p>c) The test statistics differ because a t-test is used when the population standard deviation is unknown and is estimated from the sample, while a z-test is used when the population standard deviation is known. The t-distribution is wider and has heavier tails than the z-distribution, which leads to a larger critical value and a higher chance of not rejecting the null hypothesis.</p> ####
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