In order to estimate the average daily down time,a manufacturer randomly samples 40 days of production records and found a mean of 55.576 minutes and standard deviation of 77.9 minutes.An 80% confidence interval is:
A) 55.6751 $\pm$ .28(12.32).
B) 55.6751 $\pm$ .304(12.32).
C) 55.6751 $\pm$ .28(77.9).
D) 55.6751 $\pm$ .304(77.9).

Question

In order to estimate the average daily down time,a manufacturer randomly samples 40 days of production records and found a mean of 55.576 minutes and standard deviation of 77.9 minutes.An 80% confidence interval is:
A) 55.6751 $\pm$ .28(12.32).
B) 55.6751  $\pm$ .304(12.32).
C) 55.6751 $\pm$ .28(77.9).
D) 55.6751 $\pm$ .304(77.9).

Quizplus · Accepted Answer

To find the confidence interval, we use the formula:
CI = x̄ ± (t critical)(s/√n)
where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t critical is the t-score associated with the desired confidence level and degrees of freedom (df = n-1).

Since the sample size is 40, df = 39. Using an 80% confidence level, we find the t critical value from a t-distribution table:
t critical = 1.303

Plugging in the given values, we get:
CI = 55.576 ± (1.303)(77.9/√40)
CI = 55.576 ± 12.32

Therefore, the 80% confidence interval is:
(55.576 - 12.32, 55.576 + 12.32) or (43.25, 67.90)

The closest answer choice is B,  which uses the correct sample mean and standard deviation and the correct t critical value with the formula for the confidence interval.

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