Researchers conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick, other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given in the accompanying table. Higher scores correspond to more creativily The researchers make the claim that blue enhances performance on a creative lask. "Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal Complete parts (a) and (b) Click the icon to view the summary statistics. a. Use a 0.01 significance level to test the claim that blue enhances performance on a creative task What are the null and alternative hypotheses? A. $$ H_{0}: \mu_{1} \neq \mu_{2} $$ $$ H_{1}: \mu_{1}\mu_{2} \end{array} $$ D. $$ H_{0}: \mu_{1} \geq \mu_{2} $$ $$ H_{1}: \mu_{1}

Question

Researchers conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick, other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given in the accompanying table. Higher scores correspond to more creativily The researchers make the claim that blue enhances performance on a creative lask. "Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal Complete parts (a) and (b) Click the icon to view the summary statistics. a. Use a 0.01 significance level to test the claim that blue enhances performance on a creative task What are the null and alternative hypotheses? A. $$ H_{0}: \mu_{1} \neq \mu_{2} $$ $$ H_{1}: \mu_{1}<\mu_{2} $$ B. $$ \begin{array}{l} \mathbf{H}_{0}: \mu_{1}=\mu_{2} \ \mathbf{H}_{1}: \mu_{1} \neq \mu_{2} \end{array} $$ C. $$ \begin{array}{l} \mathbf{H}_{0}: \mu_{1}=\mu_{2} \ \mathbf{H}_{1}: \mu_{1}>\mu_{2} \end{array} $$ D. $$ H_{0}: \mu_{1} \geq \mu_{2} $$ $$ H_{1}: \mu_{1}<\mu_{2} $$ The test statistic, 1, is 2.75 . (Round to two decimal places as needed.) The P-value is 0.004 , (Round to three decimal places as needed) State the conclusion for the test. A. Fail to reject the null hypothesis. There is sufficient evidence to support the claim that blue enhances performance on a creative task.

UpStudy AI · Accepted Answer

To solve this problem, we will follow the steps outlined in the prompt.

### Step 1: Extract Known Conditions
- We have two independent samples: one with a red background and one with a blue background.
- We are testing the claim that blue enhances performance on a creative task.
- The significance level is $$ \alpha = 0.01 $$.
- The test statistic is $$ t = 2.75 $$.
- The P-value is $$ 0.004 $$.

### Step 2: Formulate Hypotheses
We need to determine the null and alternative hypotheses based on the claim that blue enhances performance.

- The null hypothesis ($$ H_0 $$) typically states that there is no effect or no difference. In this case, it would state that the mean score for the blue background is equal to the mean score for the red background.
- The alternative hypothesis ($$ H_1 $$) states that there is an effect or a difference. Since we are testing if blue enhances performance, we would expect the mean score for the blue background to be greater than that for the red background.

Thus, the correct hypotheses are:
$$
\begin{array}{l}
\mathbf{H}_{0}: \mu_{1} = \mu_{2} \
\mathbf{H}_{1}: \mu_{1} > \mu_{2}
\end{array}
$$
This corresponds to option C.

### Step 3: Determine the Conclusion
Now we will compare the P-value to the significance level to make a decision about the null hypothesis.

- Given that the P-value $$ 0.004 $$ is less than the significance level $$ 0.01 $$, we reject the null hypothesis.

### Conclusion
Since we reject the null hypothesis, we conclude that there is sufficient evidence to support the claim that blue enhances performance on a creative task.

### Final Answer
- The correct hypotheses are:
$$
\begin{array}{l}
\mathbf{H}_{0}: \mu_{1} = \mu_{2} \
\mathbf{H}_{1}: \mu_{1} > \mu_{2}
\end{array}
$$
- The conclusion is: **Reject the null hypothesis. There is sufficient evidence to support the claim that blue enhances performance on a creative task.**
- **Hypotheses:** - $$ H_{0}: \mu_{1} = \mu_{2} $$ - $$ H_{1}: \mu_{1} > \mu_{2} $$ - **Conclusion:** - **Reject the null hypothesis.** - **There is sufficient evidence to support the claim that blue enhances performance on a creative task.**

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