A bank wishing to determine the average amount of time a customer must wait to be served took a random sample of 100 customers and found that the mean waiting time was 7.2 minutes. Assuming that the population standard deviation is known to be 15 minutes, find the 90% confidence interval estimate of the mean waiting time for all of the bank's customers
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To find the confidence interval estimate of the mean waiting time for all of the bank's customers, you can use the formula for a confidence interval:
Confidence Interval = Sample Mean ± (Z * (Population Standard Deviation / √n))
Where:
- Sample Mean: The mean waiting time of the random sample, which is 7.2 minutes.
- Z: The z-score corresponding to the desired confidence level. For a 90% confidence level, the z-score is 1.645.
- Population Standard Deviation: The known population standard deviation, which is 15 minutes.
- n: The sample size, which is 100 customers.
Now let's calculate the confidence interval:
Confidence Interval = 7.2 ± (1.645 * (15 / √100))
First, calculate the standard error of the mean (SEM) by dividing the population standard deviation by the square root of the sample size.
SEM = 15 / √100 = 15 / 10 = 1.5
Next, multiply the SEM by the z-score (1.645) to get the margin of error (ME).
ME = 1.5 * 1.645 = 2.4685
Finally, add and subtract the margin of error from the sample mean to get the confidence interval.
Confidence Interval = 7.2 ± 2.4685
Confidence Interval = (7.2 - 2.4685, 7.2 + 2.4685)
Confidence Interval ≈ (4.7315, 9.6685)
Therefore, with 90% confidence, the average waiting time for all of the bank's customers would fall within the range of approximately 4.7315 minutes to 9.6685 minutes.