Question

The article "National Geographic, the Doomsday Machine," which appeared in the March 1976 issue of the Journal of Irreproducible Results (yes, there really is a journal by that name -it's a spoof of technical journals!) predicted dire consequences resulting from a nationwide buildup of National Geographic magazines. The author's predictions are based on the observation that the number of subscriptions for National Geographic is on the rise and that no one ever throws away a copy of National Geographic. A key to the analysis presented in the article is the weight of an issue of the magazine. Suppose that you were assigned the task of estimating the average weight of an issue of National Geographic. How many issues should you sample to estimate the average weight to within \(0.1 \mathrm{oz}\) with \(95 \%\) confidence? Assume that \(\sigma\) is known to be 1 oz.

Short answer

The needed sample size for estimating the average weight of a National Geographic magazine within 0.1 oz with 95% confidence is 385.

Step-by-step solution

Understand the Given Values

In the exercise, you are given a few elements:\nThe confidence level, which is 95%.\nAn error margin, which is 0.1 oz.\nThe population standard deviation - denoted by the Greek letter \(\sigma\) - which is 1 oz.

Calculate the Z-value

To calculate the necessary sample size, you also need the z-value associated with the desired confidence level. The z-value is a statistical measurement that describes a value's relationship to the mean of a group of values. You can find the Z-value from a standard Z-table or alternatively use online calculators or statistical software. For a 95% confidence level, the Z-value is approximately 1.96.

Apply the Formula for Sample Size

The formula used for estimating a sample size when population standard deviation is known: \[n = \left( \frac{Z*\sigma}{E} \right)^2\] where n is the sample size, Z is the Z-value from a standard normal distribution corresponding to the desired confidence level (in this case 1.96), \( \sigma \) is the population standard deviation (1 oz) and E denotes the margin of error (0.1 oz). Plugging in these values the formula becomes: \[n = \left (\frac{1.96 * 1}{0.1}\right )^2\]

Calculate Sample Size

Carrying out the calculation will provide the required sample size. Since you can't have a fraction of a magazine, you would round up to the nearest whole number if the calculation doesn't yield an exact whole number. In this case, the formula provides a sample size of 384.16, so you would round up to 385. So, you need to sample 385 issues of National Geographic to estimate the average weight to within 0.1 oz with 95% confidence, given that the standard deviation is 1 oz.