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Question 465:



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The first thing you should do is get the mean and standard deviation of the sample of pages. You can use Excel, and use the formula =STDEV() for the standard deviation and =AVERAGE() for the mean. You should get a mean of 346.5 and standard deviation of 170.4.

First I will answer c, which will then answer a. We need a sample size which will generate a margin of error of +/-10 sq millimeters. The margin of error is calculated by multiplying the standard error of the mean (SEM) by the critical value of the t or z-distribution. As the sample size increase the values of t and z converge (esp. above 30). For part c I'll use the z-distribution then use the t-distribution on part a since the sample of 20 is relatively small. For a two-sided 99% confidence interval, the z-score is 2.58, which you lookup using the percentile to z-score calculator (enter .01).

We can then just setup and equation and solve for the unknown:

  1. The margin of error is the standard error of the mean SEM * the critical z value of  2.58, so working backwards from a margin of 10, the SEM = 10/2.58= 3.88.
  2. The SEM  of 3.88 is made up of the standard deviation divided by the square root of the needed sample size = SD/ SQRT(n). Solving for the unknown sample size n and using the standard deviation we obtained from the sample of 20. We have the equation 170.4 /sqrt(n) = 3.88
  3. Squaring both sides = 29036/n  = 15.05
  4. Isolating n gets us 15.05n= 29036 or n = 29036/15.05= 1929.3
  5. Rounding down to the nearest whole page gets us 1929.

So a huge sample size of 1929, will generate a  99% +/- 10 mm margin of error.  It should be obvious why this is a problem since the required sample is larger than the entire phone book!

Normality will likely be an issue since there are a couple extreme values (0's) in the data, which will affect the accuracy of the confidence intervals.

Lets now geneate a 95% CI around the mean, since we're using the small sample of 20, I will switch and use the t-distribution (which is better for small samples).

  1. Calculate the SEM as the SD/SQRT(n) = 170.4/4.47 = 38.1
  2. Find the critical value from the t-distribution using th Excel formula =TINV(.05,19), where 19 is the sample size minus 1 and .05 is for a 95% CI. You should get 2.09.
  3. Calculate the Margin of Error by multiplying the SEM by the critical t value = 38.1*2.09 = 79.62
  4. The end-points of the CI are the mean +/- the margin or 346.5-79.62 and 346.5+79.62 gets a 95% CI of between  266.88 and 426.12

So we can now see that a sample of 20 generates a margin of error of around 80mm for a 95% CI vs. a margin of error of 10 from one that requires a sample of 1929. A reasonable approach to me would be doubling or tripling the sample to 40-60 and using a 95% CI to get a decent idea of the ad space in the phone book.

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