The initial demand for the iMac was very high and Apple sold as many iMacs as th
ID: 3312124 • Letter: T
Question
The initial demand for the iMac was very high and Apple sold as many iMacs as they could make. One issue of great concern to Apple was whether the iMac attracting new customers and therefore expanding the base of Macintosh users or was only being bought by old customers who already owned a Macintosh. To investigate, 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 results are graphed below 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 effect of any violation on the confidence interval can be tested 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?Explanation / Answer
The objective of the study presented here is to find out the proportion of iMac buyers who are first time computer buyers.
In the study using a sample of 500 users, this proportion was 16.67% and the 95% confidence interval for this proportion is 13% to 20%.
Now, Apple has reported that one third of it iMac buyers are new computer buyers. So, the claim for this proportion by company is around 33.3%.
Margin of error ~ 3.5% ((13% + 20%)/2) - 13%
Standard error of proportion = Margin of error/Z value corresponding to confidence level
Standard error of proportion = 3.5%/1.96 = 1.78
Distance between estimated proportion by computerWare (16.7%) and claimed proportion by Apple (33.3%) in terms of standard error of proportion is (33.3% - 16.7%)/1.78 = 9 times which implies a very very less probability of such case. It appears to be sampling error (May be biased sampling) either from CompuerWare or from Apple
Before concluding on the error in sampling, few points that one has to consider.
Company has used sampling to make the claim!
How did the company make a claim on 2 million iMacs sold? Probably the company has used sampling to make this claim. If so, what is the size of the sample? and is the sampling performed across different regions as well as age groups to ensure there is no bias in sampling.
In case the company has sampled a good enough population (and used unbiased sampling techniques) to arrive at the results, then the sampling done by computerWare may be a biased one.
Factors to consider
Sample size
Demographics and other elements are taken care of to ensure no bias in sampling
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