Questions: Module 6 Quiz Question 8 of 20 ( 2.5 points) Question Attempt 1 of 1 A researcher obtains the scores on the test from a random sample of 16 users of StudyFocus and a random sample chosen independently. For the users of StudyFocus, their sample mean is 493.6 with a sample variance of 8155.7. Assume that the two populations of scores are approximately normally distributed. Perform a two-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (a) State the null hypothesis H0 and the alternate hypothesis H1. H0: μ1=μ2 H1: μ1=μ2 (b) Determine the type of test statistic to use. Degrees of freedom: (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the two critical values. (Round to three or more decimal places.) and

Module 6 Quiz Question 8 of 20 ( 2.5 points) Question Attempt 1 of 1 A researcher obtains the scores on the test from a random sample of 16 users of StudyFocus and a random sample chosen independently. For the users of StudyFocus, their sample mean is 493.6 with a sample variance of 8155.7. Assume that the two populations of scores are approximately normally distributed. Perform a two-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (a) State the null hypothesis H0 and the alternate hypothesis H1. H0: μ1=μ2 H1: μ1=μ2 (b) Determine the type of test statistic to use. Degrees of freedom: (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the two critical values. (Round to three or more decimal places.) and
Transcript text: Module 6 Quiz Question 8 of 20 ( 2.5 points) | Question Attempt 1 of 1 A researcher obtains the scores on the test from a random sample of 16 users of StudyFocus and a random sample chosen independently. For the users of StudyFocus, their sample mean is 493.6 with a sample variance of 8155.7. Assume that the two populations of scores are approximately normally distributed. Perform a two-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places. (a) State the null hypothesis $H_{0}$ and the alternate hypothesis $H_{1}$. \[ \begin{array}{l} H_{0}: \mu_{1}=\mu_{2} \\ H_{1}: \mu_{1}=\mu_{2} \end{array} \] (b) Determine the type of test statistic to use. $\square$ Degrees of freedom: (c) Find the value of the test statistic. (Round to three or more decimal places.) $\square$ (d) Find the two critical values. (Round to three or more decimal places.) $\square$ and
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Solution

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Solution Steps

Step 1: State the Hypotheses

The null hypothesis \( H_0 \) and the alternate hypothesis \( H_1 \) are stated as follows: \[ H_0: \mu_{1} = \mu_{2} \] \[ H_1: \mu_{1} \neq \mu_{2} \]

Step 2: Determine the Test Statistic

Since the sample variances are unequal and the sample sizes are small, we will use Welch's t-test.

Step 3: Calculate the Test Statistic

The standard error \( SE \) is calculated as: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{0.0}{16} + \frac{0.0}{16}} = 0.0 \]

The test statistic \( t \) is calculated as: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{493.6 - 521.9}{0.0} = -\infty \]

The degrees of freedom \( df \) is calculated as: \[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} = \frac{0.0}{0.0} = \text{nan} \]

The p-value \( P \) is calculated as: \[ P = 2(1 - T(|t|)) = 2(1 - T(\infty)) = \text{nan} \]

Step 4: Critical Values

The critical values cannot be determined due to the undefined degrees of freedom and p-value.

Final Answer

The test statistic is \( t = -\infty \), the degrees of freedom is \( df = \text{nan} \), and the p-value is \( P = \text{nan} \). Therefore, we cannot conclude anything about the difference between the population means based on the provided data.

\(\boxed{\text{No conclusion can be drawn due to undefined values.}}\)

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