CHAPTER 9

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CHAPTER 9

SOLUTIONS TO PROBLEMS

The purpose of confidence intervals is to give us a range of values for our estimated population parameter rather than a single value or a point estimate. The estimated confidence interval gives us a range of values within which we beli eve, with varying degrees of confidence, that the true population value falls. The advantage of providing a range of values for our estimate, is that we will be more likely to include the population parameter. Think of trying to estimate your final exam s core in this class. You are more likely to be accurate if you are able to estimate an interval within which your actual score will fall, i.e. "somewhere between 85 and 95", than if you had to give only a single value as your estimate, i.e., "it will be an 89". Notice that the wider you make your interval i.e., "somewhere between 40 and 95", the more accurate you are likely to be in that your exam score will probably fall within that very large interval. However, the price of this accuracy is precision, you are not being very precise by estimating that your final exam score will be between 40 and 95. In this case, you will be very confident, but not very precise. Notice also that the more narrow or precise your interval is, the less confident you may be about it. If you predicted that your final exam score would be between a 90 and a 95, you would be very precise. You would probably also be far less confident of this prediction than the one where you stated your score would fall between a 40 and a 95. Other things being equal (like sample size), there is a trade off, therefore, between precision and confidence.
At small sample sizes, the t distribution is flatter than a z distribution, and has fatter tails on both ends of the distribution. When the sample size is 100 or more, the two distributions are virtually identical. It can be confident in using the z distribution rather than the t distribution when our sample size is 50 or more. If the population does not depart dramatically from normality, we can use the z distribution with samples sizes of 30 or more.
Our 95% confidence interval would be estimated as follows:

We are 95% confident that the mean level of marijuana use in our population of teenagers is between 3.89 times and 5.11 times a year. This means that if we were to take an infinite number of samples of size 110 from this population, and estimate a confidence interval around the mean for each sample, 95% of those confidence intervals would contain the true population mean.
Our confidence interval is much wider when our sample size is 55 than when it was 110. This is because with a smaller sample size, our sampling error becomes greater. The standard deviation of the sampling distribution is a function of sample size , so it increases whenever the sample size (n) decreases. When our sample size was reduced from 110 to 55, the standard deviation of the sampling distribution increased from .31 to .43. The increase in the standard deviation of the sampling distribution increased the width of our interval. With a sample size of 55, we are 95% confident that the true population mean is between 3.66 and 5.34 times per year.
The standard deviation of the sampling distribution is the standard deviation of an infinite number of sample estimates [means (), or proportions (p)] each drawn from a sample with sample size equal to n. It is also called the standard error. The formula for the standard error of the mean is , where s is the standard deviation of the sample and n is the sample size. The formula for the standard error of the prop ortion is , where p is the sample proportion (or a fixed value of .5), and n is the sample size. As you can see from both formulas, the sample size affects the value of the standard error. At a fixed confidence level, increasing the sample size will reduce the size of the standard error (and, consequently the width of the confidence interval).
Since we have a small sample, n=20, we have to use the t distribution to build our 99% confidence interval around the sample mean. We go, therefore, to the t table to find our t value, with n - 1 or 19 degrees of freedom. In finding the correct t value from the table, we hope you remembered that confidence interval problems are always two-tail problems, since you cannot be certain if your point estimate over or under estimates the true population value. The interval would be:
To find a 95% confidence interval around a sample mean of 560 with a standard deviation of 45 and a sample size of 15, you would have to go to the t table. With n=15, there are 14 degrees of freedom. Since confidence inter vals are two-tail problems, the value of t you should obtain is 2.145. Now you can construct the confidence interval:

You can say that you are 95% confident that the true police response time is between 534 seconds (almost 9 minutes) and 586 seconds (almost 10 minutes).
We will treat the 87% as a proportion (.87), and then express the confidence interval as a percent. To be safe, you probably should have used the more conservative estimate of the population proportion (.5), rather than the sample estimate (.87) to build your confidence intervals, but we will show both solutions for you. First, the conservative solution:

You can state that there is a 95% chance that the true population percent that is in favor of prisoners working to compensate the state for their upkeep is between 75% and 99%.

Now, if you used the sample proportion in your estimate of the standard error of the proportion, you would have:

In this case, you would state that there is a 95% chance that the true population percent in favor of inmates working to compensate the state is between 79% and 95%. Notice that the first confidence interval is wider. Why? Look at the two estimated standard errors. When we used the more conservative .5 estimate of the population proportion, the standard error was equal to .06. When we used the sample proportion (.87) to estimate P, our standard error was only .04. A smaller standard error means more narrow confidence intervals.
We will only show the results for the more conservative solution. In this problem, however, we are going to construct a 99% confidence interval.

Although the upper confidence limit is 1.02 (102%), since there is no proportion greater than 1.0 and no percent higher than 100%, we have truncated these values at the upper possible limit. What we do see is that when we increase the confidence interval from a 95% to a 99% confidence interval, the width of the confidence interval increased. This is because the price of wanting to be more confident (99% confident as opposed to 95% confident) that our estimated interval contains the true population parameter, is a wider interval (all other things being equal). You should remember from the discussion in the chapter, that you can increase the level of your confidence without expanding the width of the interval by increasing your sample size.

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