(a) What is the value of the test statistic? A hypothesis test is being performed for the difference of two population means. Since the population standard deviations are known, a z test statistic will be found. The formula follows where x?1 is the mean of the sample taken from population 1, x?2 is the mean of the sample taken from population 2, D0 is the hypothesized difference between the populations, ?1 is the population standard deviation for population 1, n1 is the size of the sample taken from population 1, ?2 is the population standard deviation for population 2, n2 is the size of the sample taken from population 2. z = (x?1 - x?2 - D0) / sqrt(?1^2 / n1 + ?2^2 / n2) Recall the given data. Sample 1 Sample 2 n1 = 70 n2 = 60 x?1 = 102 x?2 = 104 ?1 = 8.8 ?2 = 7.9 Note that D0 is not in the given data, but is given with the hypotheses. Use the given hypotheses to find D0. H0: ?1 - ?2 = 0 Ha: ?1 - ?2 ? 0 D0 =
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8 σ2 = 7.9 The null hypothesis (H0) states that the difference between the two population means is 0, so D0 = 0. The formula for the z test statistic is: Show more…
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In a test of two population means with unknown variances, two independent samples of n1 = 8 and n2 = 10 were taken. The data is given below. Both populations are normally distributed: Sample From Population 1: 12, 10, 20, 20, 11, 13, 8, 9 Sample From Population 2: 20, 19, 16, 23, 17, 17, 18, 17, 10, 18 (a) You wish to test the hypothesis that both populations have the same variance. Choose the correct statistical hypotheses: Ho: σ1 = σ2 Ha: σ1 ≠ σ2 (b) Determine the value of the test statistic for this test. Use at least two decimals in your answer: Test Statistic (c) Determine the P-value for this test, to at least three decimal places. (d) Using a significance level of 0.05, the null hypothesis should be rejected if the variation in Population 1 is statistically different from the variation in Population 2.
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Step 1 (a) Find the value of the test statistic. A sample of 75 items provided a mean of 18.32. The standard deviation of the entire population is given to be 2. The hypotheses are given below. H0: μ ≥ 19 Ha: μ < 19 Recall that the test statistic, z, for a hypothesis is calculated as follows where x is the sample mean, μ0 is the hypothesized population mean, σ is the population standard deviation, and n is the sample size. z = (x - μ0) / (σ / √n) The value for μ0 is the hypothesized population mean we are testing for. Here, we have μ0 = 19. The sample of 75 items gave a mean of 18.32, so we have n = 75 and x = 18.32. Finally, the standard deviation for all items was given to be 2, so we have σ = 2. Text 2: Step 1 (a) Compute the value of the test statistic. A sample of 58 items provided a mean of x = 13 and a standard deviation of s = 6.7. The hypotheses are given below. H0: μ = 14 Ha: μ ≠ 14 A t test statistic will be used when the standard deviation is calculated from a sample instead of the population. Recall that the test statistic, t, for a hypothesis is calculated as follows where x is the sample mean, μ0 is the hypothesized population mean, s is the population standard deviation, and n is the sample size. t = (x - μ0) / (s / √n) The value for μ0 is the hypothesized population mean we are testing for. Here, we have μ0 = 14. There was a sample of 58 items given, so we have n = 58.
1. The data below come from two independent random samples from two populations. Sample 1 has 25 observations, a sample mean of 79, and a population standard deviation of 11. Sample 2 has 30 observations, a sample mean of 71, and a population standard deviation of 12. (a) Provide a 95% confidence interval for the difference between the two population means. (b) Formulate the null and alternative hypotheses to test whether there is a difference between the population means. (c) Using the critical value approach, test the hypothesis at the .05 significance level. (d) Calculate the p-value.
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