Questions: A student wants to determine if there is a difference in the pricing between two stores for health and beauty supplies. She recorded prices from both stores for each of 10 different products. Assuming that the conditions for conducting the test are satisfied, determine if there is a price difference between the two stores. Use the α=0.1 level of significance. Complete parts (a) through (d) below. A B C D E F G H I J ------------------------------ 5.92 7.46 3.78 1.75 1.78 2.81 4.76 3.13 2.98 3.74 Store 1 5.91 7.98 3.95 1.78 1.97 2.48 4.74 3.74 2.93 3.69 Store 2 C. A hypothesis test regarding two population standard deviations. D. A hypothesis test regarding the difference of two means using Welch's approximate t. (b) Determine the null and alternative hypotheses. Choose the correct answer below. A. H0: μd=0 ; H1: μd<0 B. H0: μd=0 ; H1: μd>0 C. H0: μd=0 ; H1: μd ≠ 0 D. H0: μd ≠ 0 ; H1: μd=0 (c) Use technology to calculate the P-value. (Round to three decimal places as needed.)

A student wants to determine if there is a difference in the pricing between two stores for health and beauty supplies. She recorded prices from both stores for each of 10 different products. Assuming that the conditions for conducting the test are satisfied, determine if there is a price difference between the two stores. Use the α=0.1 level of significance. Complete parts (a) through (d) below. A B C D E F G H I J ------------------------------ 5.92 7.46 3.78 1.75 1.78 2.81 4.76 3.13 2.98 3.74 Store 1 5.91 7.98 3.95 1.78 1.97 2.48 4.74 3.74 2.93 3.69 Store 2 C. A hypothesis test regarding two population standard deviations. D. A hypothesis test regarding the difference of two means using Welch's approximate t. (b) Determine the null and alternative hypotheses. Choose the correct answer below. A. H0: μd=0 ; H1: μd<0 B. H0: μd=0 ; H1: μd>0 C. H0: μd=0 ; H1: μd ≠ 0 D. H0: μd ≠ 0 ; H1: μd=0 (c) Use technology to calculate the P-value. (Round to three decimal places as needed.)
Transcript text: A student wants to determine if there is a difference in the pricing between two stores for health and beauty supplies. She recorded prices from both stores for each of 10 different products. Assuming that the conditions for conducting the test are satisfied, determine if there is a price difference between the two stores. Use the $\alpha=0.1$ level of significance. Complete parts (a) through (d) below. \begin{tabular}{|rcccccccccc|} & A & B & C & D & E & F & G & H & I & J \\ Store & 5.92 & 7.46 & 3.78 & 1.75 & 1.78 & 2.81 & 4.76 & 3.13 & 2.98 & 3.74 \\ $\mathbf{1}$ & & & & & & & & & & \\ \hline Store & 5.91 & 7.98 & 3.95 & 1.78 & 1.97 & 2.48 & 4.74 & 3.74 & 2.93 & 3.69 \\ $\mathbf{2}$ & & & & & & & & & & \end{tabular} C. A hypothesis test regarding two population standard deviations. D. A hypothesis test regarding the difference of two means using Welch's approximate $t$. (b) Determine the null and alternative hypotheses. Choose the correct answer below. A. $H_{0}: \mu_{d}=0 ; H_{1}: \mu_{d}<0$ B. $H_{0}: \mu_{d}=0 ; H_{1}: \mu_{d}>0$ C. $H_{0}: \mu_{d}=0 ; H_{1}: \mu_{d} \neq 0$ D. $H_{0}: \mu_{d} \neq 0 ; H_{1}: \mu_{d}=0$ (c) Use technology to calculate the P-value. (Round to three decimal places as needed.)
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Solution

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Solution Steps

Step 1: Null and Alternative Hypotheses

We are testing for a difference in prices between two stores for health and beauty supplies. The null and alternative hypotheses are defined as follows:

\[ H_{0}: \mu_{d} = 0 \quad \text{(no difference in prices)} \] \[ H_{1}: \mu_{d} \neq 0 \quad \text{(there is a difference in prices)} \]

Step 2: Calculate the Test Statistic

The test statistic \( t \) is calculated using the formula:

\[ t = \frac{\bar{d}}{SE} \]

Where:

  • \( \bar{d} = -0.106 \) (mean difference)
  • \( SE = 0.089 \) (standard error)

Substituting the values:

\[ t = \frac{-0.106}{0.089} = -1.193 \]

Step 3: Determine the Critical Value

For a two-tailed test at the significance level \( \alpha = 0.1 \), the critical value is determined using the t-distribution with \( df = 9 \):

\[ t_{\alpha/2, df} = t_{(0.05, 9)} = 1.833 \]

Step 4: Calculate the P-value

The P-value is calculated as follows:

\[ P = 2 \times (1 - T(|t|)) = 2 \times (1 - T(1.193)) = 0.263 \]

Final Answer

Based on the calculations:

  • The test statistic is \( t = -1.193 \).
  • The P-value is \( P = 0.263 \).
  • The critical value is \( t_{(0.05, 9)} = 1.833 \).

Since the P-value \( (0.263) \) is greater than the significance level \( \alpha = 0.1 \), we fail to reject the null hypothesis.

Thus, the final answer is:

\[ \boxed{H_{0} \text{ is not rejected; no significant price difference between the two stores.}} \]

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