Answer to Question #155484 in Statistics and Probability for EUGINE HAWEZA

Question

A sample of 150 calls to a customer helpline during one week found that callers were kept waiting on average for 16 minutes with s = 8.

(a) Find the margin of error for this result if we use a 95% confidence interval for the length of time all customers during this period are kept waiting.

(b) Interpret for management the margin of error.

(c) If we only need to be 90% confident, does the confidence interval become wider or narrower?

(d) Find the 90% confidence interval.

Answer 1

(a) Let the sample mean

$\bar{x}=16$

Let the sample standard deviation

s = 8

Let the sample size

n=150

The degrees of freedom

$d.f. = n -1 \\ = 150 -1 \\ = 149$

The t-critical value

For α/2 = 0.025 and d.f. = 149 the critical value from the t-distribution table is t = 1.976

The formula for the margin of error:

$E = t_{α/2, df}\frac{s}{\sqrt{n}} \\ E = 1.976\frac{8}{\sqrt{150}} \\ = 1.290$

Therefore, the margin of error for the result if we use a 95% confidence interval for the length of time for all customers during this period are kept waiting is 1.290.

(b) Interpretation for management the margin of error.

At 95% level the average length of time for all customers during this period are kept waiting within about 1.29 minutes of the average 16 minutes.

(c) At 90 % level, the confidence interval become narrow due to the confidence intervals width decreases, the reliability of an interval containing less if the range to possibly cover the mean.

(d) Find the 90% confidence interval for the length of time for all customers during this period are kept waiting.

Using Minitab following the steps mentioned below.

1. Import the data.

2. Select 1 Sample t from Basis Statistics option in Stat menu.

3. Enter Summarized data.

4. Click Ok.

The obtained Minitab output for the one sample t-test is

From MINITAB output, the 90% confidence interval for the length of time for all customers during this period are kept waiting is 14.92 to 17.08.

Interpretation of confidence interval:

At 90% confidence level, the length of time for all customers during this period are kept waiting is between 14.92 minutes to 17.08 minutes.