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6. The sampling distribution of the sample proportion In 2007, about 30% of new-car purchases in California 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 California can be viewed as an infinite population Define p as the proportion of the population of new-car purchases in California 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.30 You will calculate an estimate of p by drawing a simple random sample of 50 new-car purchases made in California. 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 simple random samples (of size n50) 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 .30.) Data Set Samples Sample 2 Variable Sample 2 Qualitative/ Nonnumeric Observations 50 Missing0 Values 50 Sample Filter Filtered Data Full Data Set Category include ▽ Frequency Relative ▽ Frequency Relative ▽ 0.140 0.860 1.000 Yes 7 43 50 7 43 50 0.140 0.860 1.000 Bar Chart > 0 Pie Chart Total Variable Variable Variable Correlation Correlation Suppose you select sample 2. The values of Home Equity for sample 2 are in the tool in variable Sample 2. Use the sample proportion as an estimator for the population proportion p. The estimate is Now suppose that instead of sample 2, you pull sample 12. Then your estimate of p is 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 .30. Using Minitab, 1,000 simple 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 Proportions data set.) Data Set Variable Correlation Sample Proportion Proportions Simple Linear Regression Regression Equation Sample Proportion0.00001.000 Sample Proportion Covariance 0.003954 Correlation Coefficient 1.000 Sample Proportion Sample Proportion Scatter Point Size Variable Measures Variable Units Mean 0.2997 0.2997 Standard Deviation Sample Proportion Sample Proportion 0.0629 0.0629 Crosstabulation > ANOVA: Analysis of Variance Scatter Source of Variation Sum of Squares Degrees of Freedom F Test Statisti ean Square Regression 3.950 3.950 Infinity Error 998 3.950 Correlation 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. Use the tool to sort the observed values of p from least to greatest. That is, in the Data Set section of the tool under Observations, click the downward-pointing triangle directly below the Sample Proportion label. This will sort the sample proportions.) (Hint: The worst-luck sample is the sample whose sample proportion is farthest from the true population The mean of the sampling distribution of p is sampling is . 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, one standard deviation of the mean, 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 fall within two standard deviations of the mean, and fall within three standard 140 120 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. 100 80 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 60 40 20 skewed to the right approximately normal approximately uniform neither normal nor uniform 0 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 Values of p-hat Minitab and all other trademarks and logos for the Company's products and services are the exclusive property of Minitab Inc. All other marks referenced remain the property of their respective owners. See minitab.com for more information.

Answer 1

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