an industrial designer wants to determinethe average amount of time it takes an adult to assemble an easy-to- assembletoy .use the following data (inminites)a random sample, to construct a 95% confidence interval for the mean of the population sampled:17 13 18 19 17 21 29 22 16 28 21 15 26 23 24 20 8 17 17 21 32 18 25 22 16 10 20 22 19 14 30 22 12 24 28 11
95% interval = mean ± 1.96 SD
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
To construct a 95% confidence interval for the mean of the population sampled, we can follow these steps:
Step 1: Calculate the sample mean (x̄) and the sample standard deviation (s) of the data.
Sample mean (x̄) = (17 + 13 + 18 + 19 + 17 + 21 + 29 + 22 + 16 + 28 + 21 + 15 + 26 + 23 + 24 + 20 + 8 + 17 + 17 + 21 + 32 + 18 + 25 + 22 + 16 + 10 + 20 + 22 + 19 + 14 + 30 + 22 + 12 + 24 + 28 + 11) / 35
= 621 / 35
= 17.742
Sample standard deviation (s) = √ [(Σ (xi - x̄)^2) / (n - 1)]
where Σ is the summation symbol, xi represents each data point, x̄ is the sample mean, and n is the sample size.
Substituting the values into the formula:
s = √ [( (17 - 17.742)^2 + (13 - 17.742)^2 + ... + (11 - 17.742)^2 ) / (35 - 1)]
= √ [( (-0.742)^2 + (-4.742)^2 + ... + (-6.742)^2 ) / 34]
≈ √ [(0.5502 + 22.5644 + ... + 45.29588) / 34]
≈ √ [609.6 / 34]
≈ √ [17.9529]
≈ 4.235
Step 2: Calculate the margin of error (E) using the formula:
E = t * (s / √n)
where t represents the critical value from the t-distribution table with (n - 1) degrees of freedom, s is the sample standard deviation, and n is the sample size.
For a 95% confidence interval, we determine the t-value with (35 - 1) degrees of freedom. Using a t-table or a statistical calculator, the t-value is approximately 2.03.
Substituting the values into the formula:
E = 2.03 * (4.235 / √35)
≈ 2.035 * (4.235 / 5.916)
≈ 2.035 * 0.717
≈ 1.461
Step 3: Calculate the confidence interval.
To construct the confidence interval, we add and subtract the margin of error from the sample mean.
Confidence Interval = x̄ ± E
= 17.742 ± 1.461
≈ (16.281, 19.203)
Therefore, the 95% confidence interval for the mean of the population sampled is approximately (16.281, 19.203) minutes.