Questions: 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 creativity. The researchers make the claim that "blue enhances performance on a creative task." 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) Response Summary Statistics Background μ n x̄ s Blue μ1 47 3.53 0.55 Red μ2 41 2.71 0.96 b. Construct the confidence interval suitable for testing the claim in part (a). What is it about the confidence interval that results in the same conclusion from part (a)? <μ1-μ2< (Round to two decimal places as needed.)

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 creativity. The researchers make the claim that "blue enhances performance on a creative task." 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) Response Summary Statistics Background μ n x̄ s Blue μ1 47 3.53 0.55 Red μ2 41 2.71 0.96 b. Construct the confidence interval suitable for testing the claim in part (a). What is it about the confidence interval that results in the same conclusion from part (a)? <μ1-μ2< (Round to two decimal places as needed.)
Transcript text: 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 creativity. The researchers make the claim that "blue enhances performance on a creative task." 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) Response Summary Statistics \begin{tabular}{|l|c|c|c|c|} \hline Background & $\boldsymbol{\mu}$ & $\mathbf{n}$ & $\overline{\mathbf{x}}$ & $\mathbf{s}$ \\ \hline Blue & $\boldsymbol{\mu}_{1}$ & 47 & 3.53 & 0.55 \\ \hline Red & $\boldsymbol{\mu}_{2}$ & 41 & 2.71 & 0.96 \\ \hline \end{tabular} b. Construct the confidence interval suitable for testing the claim in part (a). What is it about the confidence interval that results in the same conclusion from part (a)? $<\mu_{1}-\mu_{2}<$ (Round to two decimal places as needed.)
failed

Solution

failed
failed

Solution Steps

Step 1: Calculate the degrees of freedom

The formula for degrees of freedom when population standard deviations are not assumed to be equal is:

$df = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2}{\frac{(\frac{s_1^2}{n_1})^2}{n_1 - 1} + \frac{(\frac{s_2^2}{n_2})^2}{n_2 - 1}}$

Plugging in the values from the table:

$df = \frac{(\frac{0.55^2}{47} + \frac{0.96^2}{41})^2}{\frac{(\frac{0.55^2}{47})^2}{47 - 1} + \frac{(\frac{0.96^2}{41})^2}{41 - 1}} = \frac{(0.0064468 + 0.0225268)^2}{\frac{0.0000406}{46} + \frac{0.0005308}{40}} = \frac{0.0008362}{0.00000088 + 0.00001327} \approx \frac{0.0008362}{0.00001415} \approx 59.06 \approx 59$

Step 2: Calculate the t-score

Since we weren't given a confidence level, we'll assume a 95% confidence level. With df = 59 and α = 0.05 (two-tailed), the t-score is approximately 2.00. This value can be found using a t-table or a statistical calculator.

Step 3: Calculate the margin of error

The margin of error is calculated as:

$E = t \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = 2.00 \sqrt{\frac{0.55^2}{47} + \frac{0.96^2}{41}} = 2.00 \sqrt{0.006447 + 0.022527} \approx 2.00 \sqrt{0.028974} \approx 2.00 * 0.1702 \approx 0.340$

Step 4: Calculate the confidence interval

The confidence interval is calculated as:

$(\bar{x}_1 - \bar{x}_2) \pm E = (3.53 - 2.71) \pm 0.340 = 0.82 \pm 0.340$

So, the confidence interval is (0.48, 1.16).

Final Answer:

0.48 < μ₁ - μ₂ < 1.16

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Camila went to the food pantry with her big sister last weekend. Their mom dropped them off at 1:00 and said, "I'll pick you up at 2:30." The girls spent 5/6 of an hour sorting canned goods, 2/3 of an hour cleaning up, and 1/2 an hour reading a book to some younger children. When they went out to meet their mom, she was already there. She said, "What have you been doing? I've been waiting for you!" How long had their mom been waiting for them? Show your thinking. Express your answer as a number of minutes and as a fraction of an hour.
failedAnswered
A bag contains a red ball, a blue ball, and a white ball. Each of the three balls is drawn in turn from the bag and is not replaced. In how many different ways can the three balls be removed from the bag? A. 3 B. 4 C. 6 D. 8
failedAnswered
Which of the following is equivalent to x-1<5 ? x-1<5 5<x-1<5 -5<x-1<5 -5>x-1<5
failedAnswered
Factor. s^2 + 4s + 4 Select the correct choice below and fill in any answer boxes with your answer. A. s^2 + 4s + 4 = □ B. The polynomial is prime.
failedAnswered
Use Pythagorean theorem to find right triangle side lengths Find the value of x in the triangle shown below. Choose 1 answer: (A) x=6 (B) x=8 (C) x=sqrt(15) (D) x=sqrt(34)