11.2 POINT ESTIMATES AND CONFIDENCE INTERVALS


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1 11.2 POINT ESTIMATES AND CONFIDENCE INTERVALS Point Estimates Suppose we want to estimate the proportion of Americans who approve of the president. In the previous section we took a random sample of size 1200 from the population and used the proportion of the people in the sample who approved of the president to estimate the proportion of the people in
2 the entire country who approve of the president. The proportion of people in the sample who approve of the president is an example of what is called a point estimate. It is called a point estimate because it is a single number that estimates a population parameter, here the proportion of the people in the country who approve of the president. Example 11.3 Suppose that out of 1200 people in the sample, 615 approve of the president. The point estimate of the proportion of the people in the country who approve of the president is 615/ Example 11.4 Suppose that in the example of the 10 containers of sampled water, the sample 2 average of the 10 observed E. coli bacteria densities is 1500/cm. The point estimate 2 of the density of E. coli bacteria in the swimming pool is 1500/cm. Example 11.5 It was desired to find the population mean and standard deviation of the ages of the students enrolled at a particular community college. It was impossible to survey all the students, so a random sample of 100 students was taken and the mean and the standard deviation of the ages of the students in the sample were calculated. The sample mean was and the sample standard deviation was The sample mean age, X , is a point estimate of the population mean age,. The sample standard deviation of the ages, S 2. 25, is a point estimate of the population standard deviation,. In summary, a point estimate for a population parameter of interest is a statistic computed from the sample. It is believed to be an effective estimate of the unknown population parameter. Often, what formula to use for the statistic is obvious. For example, for the population parameter, which is the mean over a (presumably large) population, it seems that the sample mean will be a good estimate. However, in more complex situations the choice of an estimate of a population parameter is not always clear. Statisticians may find that either they have no idea how to use the sample data to estimate the population parameter of interest, or they may in fact have several equally plausible competing estimates to select from. A relatively simple example of the latter case is the estimation of 2. Let n denote the sample size. Many statisticians would use S2 sum( X X 2 n X X 2 n X i ) /, while many others would use sum( ) /( 1). Here denotes the i i
3 ith observation of the random sample. A solid argument can be made for either estimate. In fact, a significant amount of statistical theory is devoted to finding the best estimate of a population parameter. Let s consider an example in which the formula for a good estimate is not obvious. Example 11.6 Suppose the waiting time for a train that goes to the parking area from a particular terminal at Chicago s O Hare International Airport obeys a continuous uniform distribution on the time interval [0,T], where Tis an unknown population parameter. Thus a train always arrives within T minutes, but it is equally likely to arrive at any moment during this waiting time. Suppose we interview five randomly selected passengers and find that their waiting times were (in minutes) 1.4, 4.7, 4.2, 5.1, and 2.1. We want to estimate the parameter T. Clearly the sample mean is a poor choice for an estimate of T. What formula should we use? A widely used technique for producing an estimate when we have none in mind, called maximum likelihood estimation, leads us to an estimate that is equal to the maximum of the five observed times, which is 5.1 minutes. This choice of the maximum of the observed values as our point estimate is not obvious. Even though it might not be the best choice for an estimate of T, it seems clearly better than X. We will not consider maximum likelihood estimation in this book. A point estimate reports one single value that we estimate to be the true value of the population parameter. However, point estimates have the important limitation of not informing us how much the estimate is likely to be in error. Whenever we estimate a population parameter, we lack total accuracy. Thus our estimate will almost always be different from the actual population parameter. So the estimate will almost always have some amount of error associated with it. By simply reporting the point estimate of a parameter, we have essentially ignored the important issue of the likely error size associated with the estimate. Example 11.7 Reconsider Example A second random sample, this time of 200 students, was taken. Suppose, to keep our explanation simple, the point estimate of the population mean is the same in both cases, namely The standard deviation of the second sample mean would be much lower than the standard deviation of the first mean, because the second sample is larger. So even though the two point estimates of the population mean are the same, the second one is surely more accurate, because it is based on more information namely, twice as many observations.
4 Confidence Intervals To improve on point estimates, statisticians usually report an interval of values that they believe the parameter is highly likely to lie in. Usually the point estimate is the middle point of the interval and the endpoints of the interval communicate the size of the error associated with the estimate (recall that point estimates ignore this error) and how confident we are that the population parameter is in the interval. The intervals are called confidence intervals. Typical confidence levels used in practice for confidence intervals are 90%, 95%, or 99%, with 95% occurring most frequently in applications. In Examples 11.1 and 11.3, if we are given a 95% confidence interval for the proportion of the population that approves of the president, which can be shown to be the interval (0.48, 0.54), we say we are 95% confident that the population proportion is contained in the confidence interval. In Examples 11.2 and 11.4, given a 90% confidence interval for the density of bacteria in the swimming pool, (1490, 1510) say, we say we are 90% confident that the density of bacteria in the swimming pool is contained in the confidence interval. (We will learn how to calculate confidence intervals for different population parameters later in the chapter.) What does it really mean to state a 95% confidence interval for the unknown population proportion approving of the president? Although 95% sounds impressive, we cannot be satisfied unless we understand what it means. Theoretically, it means that the probability is 0.95 that such a confidence interval, which will be random because the sample it is formed from is random, will contain (surround) the unknown proportion in the population approving of the president. Our experimental view of probability based on the fivestep method will help us more clearly and deeply understand what this probability of 0.95 means practically. Just as we do simulations over and over in the fivestep method, imagine that a statistician does the sampling experiment of Example 11.1 over and over 1000 times, say and each time computes a 95% confidence interval from the 1200 sampled people. Now we can find the experimental probability given by the proportion of the 1000 confidence intervals that actually covers the true fraction of the population favoring the president. Since 0.95 is the theoretical probability of the interval containing the population proportion, this experimental probability of the 1000 confidence intervals including the true value will also be close to (Below, we will simulate 100 such 95% confidence intervals and calculate the experimental confidence interval probability.) Of course, in a real application there will only be one random sample and hence only one such confidence interval, such as the (0.48, 0.54) interval of Examples 11.1 and But the statistician obtaining this one
5 sample knows, because of the experimental probability viewpoint, that this confidence interval is very likely to be correct in the sense that it contains the true value (since about 95% of such confidence intervals would cover the true population proportion). In the case of the (0.48, 0.54) interval of Examples 11.1 and 11.3, we know it is very likely that the true proportion of people favoring the president lies between 0.48 and Now in light of this insight into how to interpret the confidence interval percentage, let s return to the Key Problem. The St. Louis Post Dispatch explained the concept of its reported confidence interval this way: [A 95% confidence interval] means if the survey were taken 100 times, the results for the [random] group of respondents would each vary no more than 5.7 percent in either direction from the true population percentage opposing the stadium [about] 95% of these times. The Post Dispatch quote is a bit roundabout and hence forces us to go through a slightly tricky piece of logic (draw yourself a picture if needed). If the interval, which extends 5.7% in either direction from its midpoint, indeed varies no more than 5.7% in either direction from the true population mean about 95% of the time, then about 95% of these intervals must contain the true population percentage as desired. The Post Dispatch could have more simply and more directly told its readers that such an interval can be expected to contain the true population parameter about 95% of the time. In other words, just as explained above in the presidential popularity example, the Post Dispatch is pointing out that if you take 100 random samples from the same population and calculate the confidence interval for the population proportion for each sample, about 95% of the confidence intervals will include the true population proportion. That is, you will be correct in your claim that the unknown population proportion is in the interval computed using the sample for about 95 of the 100 samples. Using our fivestep method, we now simulate 100 confidence intervals and determine how many of them contain the true population parameter. Suppose that for the Key Problem the true proportion of people in the county who are opposed to the stadium is 50% (remember that the parameter value is never known to the statistician). In that case the probability that a person in the sample will be opposed to the stadium is the same as the probability of heads seen in flipping a fair coin: 0.5. Our goal is to obtain 100 samples of 301 people and calculate the confidence interval for each of these 100 samples (we will learn to compute such confidence intervals in Section 11.6 below). Each sampling is the same as flipping a coin 301 times, recording the number of heads, and calculating the confidence interval for the proportion of heads in the sample. We then repeat the process 100 times. The 100 confidence intervals obtained from this process are represented in Figure 11.1.
6 Figure 11.1 One hundred simulated 95% confidence intervals for the Key Problem assuming a 50/50 population split. The line across the middle of the graph represents the true population proportion, The 100 confidence intervals are the vertical lines on the graph. If the confidence interval covers the 0.50 line, then we say that the true population proportion is contained in the interval. Likewise, if the confidence interval does not cover the 0.50 line, then we say that the true population proportion is not contained in the interval. For the graph of Figure 11.1, we see that 96 out of the 100 confidence intervals cover the 0.50 line. So for the 100 confidence intervals, 96% of them (about 95%, as expected) contain the true population parameter, Thus, this example of the fivestep method clearly illustrates how we are to correctly interpret a 95% confidence interval. As already discussed above, it is a sort of statisticians success rate or batting average. If a statistician constructs % confidence intervals during a year s work, then, as our fivestep simulation confirms, we can expect about 95% of them to be hits : cases in which the population parameter is contained in the interval. Similarly we would expect about 5% of them to be outs, or misses: cases in which the population parameter is not contained in the interval. Compared with baseball, in which a batting average of is considered great, a confidence interval coverage rate for statisticians is what is usually required. Batting is never guaranteed in baseball, but in fact a statistician can guarantee a 95% confidence interval, as we shall see. Confidence intervals have two basic characteristics that we need to understand. First, given the same set of data, a 95% confidence interval is wider than a 90% confidence interval, and a 99% confidence interval is wider than a 95% confidence interval. Thus, the higher the confidence level we require, the wider the interval we are forced to accept! Of course, a very wide interval is of little use to the scientist who has sought statistical advice. Thus there is no free lunch in specifying a 99% confidence instead of a
7 SECTION 11.2 EXERCISES 95% confidence, because the price paid is a wider interval. Here are the 90%, 95%, and 99% confidence intervals for the proportion of people in the country who approve of the president in Example 11.1: 90%: (0.49, 0.53) 95%: (0.48, 0.54) 99%: (0.47, 0.55) As you can see from this example, the higher the level of confidence, the wider the confidence interval needs to be in order to contain the population proportion with the specified confidence. The second characteristic of confidence intervals is that, given the same confidence level, a shorter and hence more informative confidence interval is associated with more data points. Suppose in the situation of Example 11.1 a sample of 2400 people was taken from the population and the number of people in the sample that approved of the president was The point estimate of the population proportion would be the same: 1230/ However, a 95% confidence interval for the population proportion based on this sample of 2400 can be shown to be (0.4925, ). This is shorter than the 95% confidence interval for the original sample of 1200, which can be shown to be (0.4842, ). Indeed, the result of increasing the sample size is a shorter interval (of length 0.04 compared with 0.057) for the same confidence level of 95%. In summary, point estimates provide only a single number to estimate the value of a population parameter. By contrast, confidence intervals give a range of values that we reasonably expect will contain the population parameter. Again, a 95% confidence interval means that if we were to take a large number of samples (like 100 or 1000) of equal size from the same population and calculate a confidence interval for the population parameter, about 95% of the confidence intervals would contain the true value of the population parameter. 1. Suppose we want to estimate the proportion 3. Explain the meaning of a 99% confidence of a city s residents who drive to work. What level. is a good choice for the point estimate of this 4. Suppose you want a confidence interval for proportion? a population proportion. You want to be as 2. Suppose, instead, we want to estimate the av accurate as possible, so you select a 100% conerage number of miles people living in a city fidence level. What would your confidence drive to work. What are two possible choices interval have to be? for the point estimate of this average?
8 5. Which confidence interval, when based on 7. A Gallup poll of 1013 adults found 61% of the same data, is wider: an 80% or an 85% the people in the sample drink alcoholic bevconfidence interval? erages, which yields a confidence interval of 6. True or false: If you flip a fair coin 100 times, (58%, 64%). True or false: There is an apthen calculate a 95% confidence interval, there proximate 95% chance that the percentage of is an approximate 95% chance 1/2 will be in adults in the population who drink alcoholic the interval. beverages is between 58% and 64%.
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