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# Confidence Interval and Sampling Error

ComputerWare asked 500 of its iMac purchasers what computer they owned or had owned. There were 83 new computer owners, 60 who had owned a Windows-based computer, and 357 who had owned a Macintosh. The proportion of first-time computer owners was 83/500 = 0.167. So, 0.167 is an estimate of the proportion of new computer owners. To determine the margin of error, compute the 95% confidence interval on the proportion. It extends from 0.13 to 0.20. Therefore, it is likely that between 13% and 20% of iMac buyers are first-time computer owners.

Assumptions: The confidence interval on the proportion assumes that the sampling distribution of p is normal. The sampling distribution with N = 500 and the true portion (Pi) being 0.1667 can be found using the "Normal Approximation to the Binomial" demonstration. The effect of any violation on the confidence interval can be tested using the "Confidence interval on Pi" demonstration.

A second assumption is that each observation(customer) is independent of each other observation. There is no reason to suppose this assumption is violated.

Epilog: After one year and the sale of 2,000,000 iMacs, Apple reports that one third of the new buyers of iMacs are first-time computer buyers. This is outside the 95% confidence interval that runs extends from 0.13 to 0.20. Is this sampling error or might there be other factors at work here?

#### Solution Preview

Please see response attached, also presented below. I hope this helps and take care.

RESPONSE:

Epilog: After one year and the sale of 2,000,000 iMacs, Apple reports that one third of the new buyers of iMacs are first-time computer buyers. This is outside the 95% confidence interval that runs extends from 0.13 to 0.20. Is this sampling error or might there be other factors at work here?

Time and changing buying trends could be potential factors at work here. Since one year has passed the computer trends may have changed over time, so 33.33% could indeed be a true reflection of this population of iMacs users at this point in time. There are some other considerations as well.

Factors that Affect Confidence Intervals

There are three factors that determine the size of the confidence interval for a given confidence level. These are: sample size, percentage and population size.

1. Sample ...

#### Solution Summary

Based on the case provided, this solution discusses the factors that might help explain why Apple reports that one third of the new buyers of iMacs are first-time computer buyers, which falls outside the previous estimate a year ago using a 95% confidence interval.

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