Question

Show your solutions using PhStat only

1. The manager of a bank that has 1,000 depositors wants you to estimate the proportion of its depositors with more than one account at the bank. A random sample of 100 depositors is selected, and 30 state that they have more than one account at the bank.
2. 1. Construct a 90% confidence interval estimate of the population proportion of the bank’s depositors who have more than one account at the bank. Interpret the interval estimate.
3. 2. Given the margin of error in (a), what sample size is needed to estimate the population proportion to within with 90% confidence?

An automobile shop manager timed six employees and found that the average time it took them to change a water pump was 18 minutes and he standard deviation was 3 minutes. Assuming a normal population, find the 99% confidence interval of the true mean. Interpret the interval estimate.

A recent study of 28 city residents showed that the mean of the time they had lived at their present address was 9.3 years and the standard deviation was 2 years. Assuming a normal population, find the 90% confidence interval of the true mean. Interpret the interval estimate.1 1.

Answer 1

Part a

The required PhStat output for the confidence interval is given as below:

Confidence Interval Estimate for the Proportion
Data
Sample Size	100
Number of Successes	30
Confidence Level	90%
Intermediate Calculations
Sample Proportion	0.3
Z Value	-1.6449
Standard Error of the Proportion	0.0458
Interval Half Width	0.0754
Confidence Interval
Interval Lower Limit	0.2246
Interval Upper Limit	0.3754

Interpretation: We are 90% confident that the true population proportion of the bank’s depositors who have more than one account at the bank will lies between 22.46% to 37.54%.

Part b

The required PhStat output is given as below:

Sample Size Determination
Data
Estimate of True Proportion	0.3
Sampling Error	0.0458
Confidence Level	90%
Intermediate Calculations
Z Value	-1.6449
Calculated Sample Size	270.8587
Result
Sample Size Needed	271

Part c

Required output is given as below:

Confidence Interval Estimate for the Mean
Data
Sample Standard Deviation	3
Sample Mean	18
Sample Size	6
Confidence Level	99%
Intermediate Calculations
Standard Error of the Mean	1.224744871
Degrees of Freedom	5
t Value	4.0321
Interval Half Width	4.9383
Confidence Interval
Interval Lower Limit	13.06
Interval Upper Limit	22.94

We are 99% confident that the average time it took them to change a water pump will lies between 13.06 minutes and 22.94 minutes.

Part d

The required output is given as below:

Confidence Interval Estimate for the Mean
Data
Sample Standard Deviation	2
Sample Mean	9.3
Sample Size	28
Confidence Level	90%
Intermediate Calculations
Standard Error of the Mean	0.377964473
Degrees of Freedom	27
t Value	1.7033
Interval Half Width	0.6438
Confidence Interval
Interval Lower Limit	8.66
Interval Upper Limit	9.94

We are 90% confident that the true population mean time they had lived at their present address will lies between 8.66 and 9.94.