In a randomized comparative experiment on the effect of color on the performance of a cognitive task, researchers randomly divided 69 subjects (27 males and 42 females ranging in age from 17-25 years) into three groups. Participants were asked to solve a series of six anagrams. One group was presented with the anagrams on a blue screen, one group saw them on a red screen, and one group had a neutral screen. The time, in seconds, taken to solve the anagrams was recorded. The paper reporting the study gives the sample mean X = 11.58 and sample standard deviation s = 4.37 for the times of the 23 members of the neutral group. Compute the 95% confidence interval for the mean time in the population from which the subjects were recruited. (Table A6 is given on the next page.) Relevant formula: x +t Et

Question

24-03-2025
Mathematics

College

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In a randomized comparative experiment on the effect of color on the performance of a cognitive task, researchers randomly divided 69 subjects (27 males and 42 females ranging in age from 17-25 years) into three groups. Participants were asked to solve a series of six anagrams. One group was presented with the anagrams on a blue screen, one group saw them on a red screen, and one group had a neutral screen. The time, in seconds, taken to solve the anagrams was recorded. The paper reporting the study gives the sample mean X = 11.58 and sample standard deviation s = 4.37 for the times of the 23 members of the neutral group. Compute the 95% confidence interval for the mean time in the population from which the subjects were recruited. (Table A6 is given on the next page.) Relevant formula: x +t Et

Answer :

Final answer:

The 95% confidence interval for the mean time in the population is approximately 9.687 to 13.473 seconds.

Explanation:

To compute the 95% confidence interval for the mean time in the population, we need to use the formula:

x ± t * (s / √n)

Given that the sample mean (x) for the neutral group is 11.58 and the sample standard deviation (s) is 4.37, we can substitute these values into the formula.

First, we need to find the critical value (t) from the t-distribution table. The degrees of freedom for this calculation is the sample size minus one, which is 23 - 1 = 22. Looking up the critical value for a 95% confidence level and 22 degrees of freedom in the t-distribution table, we find that t = 2.074.

Now we can substitute the values into the formula:

11.58 ± 2.074 * (4.37 / √23)

Calculating the expression, we get:

11.58 ± 2.074 * 0.913

Simplifying further, we have:

11.58 ± 1.893

Therefore, the 95% confidence interval for the mean time in the population is approximately 9.687 to 13.473 seconds.

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