5. The sampling distribution of the sample proportion Aa Aa In 2007, about 20% of new-car purchases in Florida were financed with a home equity loan. [Source : "Auto Industry Feels the Pain of Tight Credit," The New York Times, May 27, 2008.] The ongoing process of new-car purchases in Florida can be viewed as an infinite population. Define p as the proportion of the population of new-car purchases in Florida that are financed with a home equity loan. The true population value of p is not known, but for the sake of this exercise assume that p.20 You will calculate an estimate of p by drawing a random sample of 50 new-car purchases made in Florida. If the purchase was financed with a home equity loan, you record a value of Yes for the variable Home Equity. If the purchase was not financed with a home equity loan, you record a value of No for the variable Home Equity Since the population is infinite, an infinite number of samples can be drawn from the population. The sample data for the variable Home Equity for 15 random samples (of size n = 50) that could be pulled from the population are in the data set samples. (Each value of Home Equity is a random selection from a binomial population with p .20.) Simple random samples (n 50) drawn from a binomial population (p0.20 Minitab was used to generate the samples. Data Set Samples Sample Observations ariables 15 Observations 50 Type Variablev Form V Values 50 50 50 50 50 50 50 50 50 50 50 50 Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Sample 7 Sample 8 Sample 9 Sample10 Sample 11 Sample 12 Qualitative Nonnumeric Qualitative Nonnumeric Qualitative Nonnumeric Qualitative Nonnumeric QualitativeNonnumeri Qualitative Nonnumeric QualitativeNonnumeri Qualitative Nonnumeric Qualitatve Nonnumeric Qualitative Nonnumeric Qualitative Nonnumeric Qualitative Nonnumeric Variables Observations > Variable Variable Suppose you select sample 3. The values of Home Equity for sample 3 are in the tool in variable Sample 3. Use the sample proportion as an estimator for the population proportion p. The point estimate is 26 Now suppose that instead of sample 3, you pull sample 9. Then your point estimate of p is .12 Selecting a random sample is an example of a statistical experiment, and the sample statistic p is a numerical description of the result of the experiment. Therefore, p is a random variable. The probability distribution of p is called the sampling distribution of p In practice, you select one random sample and use the information from that sample to estimate the population parameter of interest. However, statisticians sometimes perform a procedure called repeated sampling, in which the experiment is run over and over again, and the value of the sample statistic from each run of the experiment is recorded. The distribution of the sample statistics from the repeated sampling is an approximation of the sample statistic's sampling distribution. Repeated sampling is used to develop an approximate sampling distribution for p when n = 50 and the population from which you are sampling is binomial with p .20. Using Minitab, 1,000 random samples are drawn. The sample proportions for the 1,000 samples are located in the Proportions data set in the variable Sample Proportion. (The 15 samples in the Samples data set correspond to the first 15 sample proportions in the Pronortions data set If the sample you select for your statistical study is one of the 1,000 samples we drew in our repeated sampling, the worst-luck sample you could draw is proportion. Us click the dowr sample 794 (Hint: The worst-luck sample is the sample whose sample proportion is farthest from the true population the observed values of p from least to greatest. That is, in the Data Set section of the tool under Observations, ngle directly below the Sample Proportion label. This will sort the sample proportions.) sample 132 sample The mean of the sampling distribution of p is sampling is .20 . The mean of the approximation of the sampling distribution obtained from repeated The standard deviation of the sampling distribution of p is obtained from repeated sampling is . The standard deviation of the approximation of the sampling distribution Consider the the approximation of the sampling distribution obtained from repeated sampling. Of the sample proportions, 61.4% fall within one standard deviation of the mean, 95.0% fall within two standard deviations of the mean, and deviations of the mean. [Hint: Use the tool to filter for the desired sample proportions. That is, under the filter section of the Statistics page for the variable Sample Proportion, enter the appropriate values for Minimum and Maximum in their respective entry fields. Applying the filter will then allow you to obtain the number of values (located directly across from the Values label) that lie within the specified range.] fall within three standard 160 140 120 100 80 60 40 20 Examine the frequency histogram for the the approximation of the sampling distribution obtained from repeated sampling shown at right. A normal curve is superimposed on the histogram Based on the histogram and your answers to the previous set of questions, the the approximation of the sampling distribution obtained from repeated sampling appears to be