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Inference about a Population Proportion BPS chapter 19 © 2010 W.H. Freeman and Company Sample proportion Which of the following symbols represents the sample proportion? a) b) c) d) P-value Sample proportion (answer) Which of the following symbols represents the sample proportion? a) b) c) d) P-value Sampling distribution The theoretical sampling distribution of a) b) c) d) p̂ Gives the values of p̂ from all possible samples of size n from the same population. Provides information about the shape, center, and spread of the values in a single sample. Can only be constructed from the results of a single random sample of size n. Is another name for the histogram of the values in a random sample of size n. Sampling distribution (answer) The theoretical sampling distribution of a) b) c) d) p̂ Gives the values of p̂ from all possible samples of size n from the same population. Provides information about the shape, center, and spread of the values in a single sample. Can only be constructed from the results of a single random sample of size n. Is another name for the histogram of the values in a random sample of size n. Sampling distribution Which of the following is true? a) b) c) The shape of the sampling distribution of p̂ is always bell-shaped. The shape of the sampling distribution of p̂ gets closer to the shape of the population distribution as n gets large. The shape of the sampling distribution of p̂ becomes approximately normal as n gets large. Sampling distribution (answer) Which of the following is true? a) b) c) The shape of the sampling distribution of p̂ is always bell-shaped. The shape of the sampling distribution of p̂ gets closer to the shape of the population distribution as n gets large. The shape of the sampling distribution of p̂ becomes approximately normal as n gets large. Sampling distribution Which of the following best describes the mean of the sampling distribution of p̂ ? a) b) c) As n increases, the mean of the sampling distribution of p̂ gets closer to p. As n increases, the mean of the sampling distribution of p̂ gets closer to x . Regardless of the value of n, the mean of the sampling distribution of p̂ is equal to p. Sampling distribution (answer) Which of the following best describes the mean of the sampling distribution of p̂ ? a) b) c) As n increases, the mean of the sampling distribution of p̂ gets closer to p. As n increases, the mean of the sampling distribution of p̂ gets closer to x . Regardless of the value of n, the mean of the sampling distribution of p̂ is equal to p. Sampling distribution True or false: As n increases, the standard deviation of the sampling distribution of p̂ gets smaller. a) b) True False Sampling distribution (answer) True or false: As n increases, the standard deviation of the sampling distribution of p̂ gets smaller. a) b) True False Sampling distribution Ten percent of all customers of Cheap Foods regularly purchase GoodEnuf Brand Chicken Fingers. We plan to ask a random sample of 45 Cheap Foods customers if they regularly purchase Good-Enuf Chicken Fingers. We will then calculate p̂ from the responses. Is the shape of the sampling distribution of p̂ close enough to normal to use the normal distribution to compute probabilities on p̂ ? a) b) c) d) Yes, because n > 30. No, because np = (45) (0.10) = 4.5 which is < 10. No, because we only have data from one sample. We cannot know the shape without knowing how many of the 45 customers purchase Good-Enuf Chicken Fingers. Sampling distribution (answer) Ten percent of all customers of Cheap Foods regularly purchase GoodEnuf Brand Chicken Fingers. We plan to ask a random sample of 45 Cheap Foods customers if they regularly purchase Good-Enuf Chicken Fingers. We will then calculate p̂ from the responses. Is the shape of the sampling distribution of p̂ close enough to normal to use the normal distribution to compute probabilities on p̂ ? a) b) c) d) Yes, because n > 30. No, because np = (45) (0.10) = 4.5 which is < 10. No, because we only have data from one sample. We cannot know the shape without knowing how many of the 45 customers purchase Good-Enuf Chicken Fingers. Sampling distribution A potential candidate for president has stated that she will run for office if at least 30% of Americans voice support for her candidacy. To make her decision, she draws a random sample of 500 Americans. Suppose that in fact 35% of all Americans support her candidacy. The mean for the sampling distribution of p̂ is a) 0.30 b) 0.35 c) Cannot be determined without more information. Sampling distribution (answer) A potential candidate for president has stated that she will run for office if at least 30% of Americans voice support for her candidacy. To make her decision, she draws a random sample of 500 Americans. Suppose that in fact 35% of all Americans support her candidacy. The mean for the sampling distribution of p̂ is a) 0.30 b) 0.35 c) Cannot be determined without more information. Confidence interval The purpose of a confidence interval for p is a) b) c) d) To give a range of reasonable values for the level of confidence. To give a range of reasonable values for the sample proportion. To give a range of reasonable values for the population proportion. To give a range of reasonable values for the difference between the sample proportion and the population proportion. Confidence interval (answer) The purpose of a confidence interval for p is a) b) c) d) To give a range of reasonable values for the level of confidence. To give a range of reasonable values for the sample proportion. To give a range of reasonable values for the population proportion. To give a range of reasonable values for the difference between the sample proportion and the population proportion. Confidence interval The confidence interval formula for p does NOT include a) b) c) d) e) The sample proportion. The z* value for specified level of confidence. The margin of error. The sample size. The population size. Confidence interval (answer) The confidence interval formula for p does NOT include a) b) c) d) e) The sample proportion. The z* value for specified level of confidence. The margin of error. The sample size. The population size. Standard error The standard error a) b) c) Is the true standard deviation of the sampling distribution of p̂ . Is an estimate, using sample data, of the standard deviation of the sampling distribution of p̂ . Measures the maximum difference expected between p and p̂ at a specified level of confidence. Standard error (answer) The standard error a) b) c) Is the true standard deviation of the sampling distribution of p̂ . Is an estimate, using sample data, of the standard deviation of the sampling distribution of p̂ . Measures the maximum difference expected between p and p̂ at a specified level of confidence. Confidence interval In 1993, presidential candidate Ross Perot appeared on television to voice his opinions on government reform. To gauge public opinion, Perot urged viewers to fill out the survey appearing in that week’s issue of TV Guide. Of the approximately 1.4 million respondents, 98% agreed with Perot’s platform on health care reform. Are the assumptions met for computing a confidence interval for the proportion of U.S. adults that agree with Ross Perot’s health care reform platform? a) b) c) d) Yes, because the sample size is large. Yes, because the population is at least 10 times larger than the sample. Both of the above answers are correct. No, because the data are from a voluntary response sample. Confidence interval (answer) In 1993, presidential candidate Ross Perot appeared on television to voice his opinions on government reform. To gauge public opinion, Perot urged viewers to fill out the survey appearing in that week’s issue of TV Guide. Of the approximately 1.4 million respondents, 98% agreed with Perot’s platform on health care reform. Are the assumptions met for computing a confidence interval for the proportion of U.S. adults that agree with Ross Perot’s health care reform platform? a) b) c) d) Yes, because the sample size is large. Yes, because the population is at least 10 times larger than the sample. Both of the above answers are correct. No, because the data are from a voluntary response sample. Margin of error The margin of error covers a) b) c) d) Random sampling error. Undercoverage error. Non-response error. All of the above. Margin of error (answer) The margin of error covers a) b) c) d) Random sampling error. Undercoverage error. Non-response error. All of the above. Margin of error When an opinion poll states with 95% confidence the margin of error for the sample percentage is plus or minus 3 percentage points, this means that a) b) c) d) Between 92% and 98% of the people chosen for the sample were contacted. The percentage of people who said “Yes” to the question was between 92% and 98%. 95% of the time, the sample percentage differs from the true population value by exactly 3 percentage points. 95% of all samples chosen using the same method will give a sample percent within 3 percentage points of the true population value. Margin of error (answer) When an opinion poll states with 95% confidence the margin of error for the sample percentage is plus or minus 3 percentage points, this means that a) b) c) d) Between 92% and 98% of the people chosen for the sample were contacted. The percentage of people who said “Yes” to the question was between 92% and 98%. 95% of the time, the sample percentage differs from the true population value by exactly 3 percentage points. 95% of all samples chosen using the same method will give a sample percent within 3 percentage points of the true population value. Hypotheses Suppose we are interested in testing H 0 : p p0 H a : p p0 What is a) b) c) p0? The sample proportion. The population proportion. The hypothesized value of the population proportion. Hypotheses (answer) Suppose we are interested in testing H 0 : p p0 H a : p p0 What is a) b) c) p0? The sample proportion. The population proportion. The hypothesized value of the population proportion. P-value For a test of hypotheses for p, the P-value is a) b) c) The probability that p equals p0. The probability of getting a p̂ equal to our observed value of p̂ or more extreme computed assuming the null hypothesis is true. The probability of getting a p̂ equal to our observed value of p̂ or more extreme computed assuming the alternative hypothesis is true. P-value (answer) For a test of hypotheses for p, the P-value is a) b) c) The probability that p equals p0. The probability of getting a p̂ equal to our observed value of p̂ or more extreme computed assuming the null hypothesis is true. The probability of getting a p̂ equal to our observed value of p̂ or more extreme computed assuming the alternative hypothesis is true. Margin of error If you increase the acceptable margin of error from 0.2 to 0.3, the required sample size will a) b) c) d) Decrease. Remain the same. Increase. Either increase or decrease because the sample sizes vary according to chance. Margin of error (answer) If you increase the acceptable margin of error from 0.2 to 0.3, the required sample size will a) b) c) d) Decrease. Remain the same. Increase. Either increase or decrease because the sample sizes vary according to chance. Margin of error Suppose you want to estimate the proportion of adults in Vermont (population 0.6 million) that approve of the new health care bill. You also want to estimate the proportion of adults in New York (population 19 million) that approve the new health care bill. To achieve the same margin of error in a 95% confidence interval for p in Vermont and New York, the required sample size of the Vermont sample will be a) b) c) Smaller than the New York sample. Larger than the New York sample. Approximately the same as the New York sample. Margin of error (answer) Suppose you want to estimate the proportion of adults in Vermont (population 0.6 million) that approve of the new health care bill. You also want to estimate the proportion of adults in New York (population 19 million) that approve the new health care bill. To achieve the same margin of error in a 95% confidence interval for p in Vermont and New York, the required sample size of the Vermont sample will be a) b) c) Smaller than the New York sample. Larger than the New York sample. Approximately the same as the New York sample. Non-response You sent out the number of surveys required by the sample size formula for an acceptable margin of error and your non-response rate is 50%. What affect will this have on your results? a) b) c) d) Decrease the margin of error, but not affect the accuracy of the results. Increase the margin of error, but not affect the accuracy of the results. Bias the results if the non-response is due to some characteristic of the population. Bias the results and decrease the margin of error. Non-response (answer) You sent out the number of surveys required by the sample size formula for an acceptable margin of error and your non-response rate is 50%. What affect will this have on your results? a) b) c) d) Decrease the margin of error, but not affect the accuracy of the results. Increase the margin of error, but not affect the accuracy of the results. Bias the results if the non-response is due to some characteristic of the population. Bias the results and decrease the margin of error.