The interval is called a confidence interval and the spread of the interval is dependent in part on the spread of the sample statistic around the true population parameter. We know how sample proportions vary around the true population proportion from last lesson when we introduced the sampling distribution of a sample proportion. The spread of our sampling distribution depended on the population proportion value (p) and the sample size (n) (i.e. the standard deviation of the sampling distribution was derived using a formula that used the population proportion and sample size). However, now we do not know the population proportion (if we did, then we would not have to use our sample value to make an inference about the population value). We substitute the sample proportion value into our formula for the standard deviation of the sampling distribution and now call this measure of spread the standard error of the sample proportion. Standard Error of the Sample Proportion (SEP) = estimated standard deviation of the sample proportion sampling distribution = (sample proportion*(1-sample proportion)/sample size) 1. A random sample of 100 freshmen students at University Park campus is taken to estimate the percentage of freshmen that have taken a tour of Pattee Library during orientation week. Of the 100 freshmen, 40 (40%) of them have taken a tour of Pattee Library. The standard error (SE) for a sample percentage in this situation is estimated to be approximately: A) 4.0% B) 4.9% C) 0.24% D) 24% 2. A random sample of 400 freshmen students at University Park campus is taken to estimate the percentage of freshmen who have taken a tour of Pattee Library during orientation week. Of the 400 freshmen, 160 (40%) of them have taken a tour of Pattee Library. The standard error (SE) for a sample percentage in this situation is estimated to be approximately: A) 4.9% B) 4.0% C) 2.45% D) 24% 3. When the sample size increases by 4 times (going from a sample of size 100 to a sample of size 400 with everything else remaining the same), the SE of the sampling distribution for our sample proportion is: A) as large as it was before the sample size increase B) 2 times larger than it was before the sample size increase C) 4 times larger than it was before the sample size increase D) as large as it was before the sample size increase 4. As sample size increases: A) the SE increases B) the SE remains the same C) the SE decreases Use the information below to answer the questions that follow: The confidence interval for our true population proportion (for large random samples) is given by: sample proportion +/- z*SEP Where z* is a multiplier of the SEP and the z value used depends on the level of confidence (see Table 10.1 in the online notes for z values for a particular level of confidence). 5. A random sample of 400 freshmen students at University Park campus is taken to estimate the percentage of freshmen who have taken a tour of Pattee Library during orientation week. Of the 400 freshmen, 160 (40%) of them have taken a tour of Pattee Library. The 95% confidence interval for the population percentage for all freshmen who have taken a tour of Pattee Library during orientation week is: A) 40% +/- 2.45% B) 40% +/- 4.90% C) 95% +/- 40% D) 5% +/- 40% 6. An exit poll was done on election night. The exit poll found that 204 out of 400 randomly selected voters (51%) had voted for the incumbent Senator. The 95% confidence interval for the population percentage of all voters who voted for the incumbent Senator in that election is: A) 51% +/- 5% B) 51% +/- 2.5% C) 95% +/- 5% D) 95% +/- 2.5% 7. A random sample of 225 residents of a small mid-western town found that 80% approve of a new parking ordinance. The 95% confidence interval for the true population percentage of all residents who favor the parking ordinance is: A) 80% +/- 5.33% B) 80% +/- 2.666% C) 80% +/- 8% D) 80% +/- 2% 8. Based on an exit poll done on election night, the 95% confidence interval for the population percentage of all voters who voted for the incumbent Senator was 52% +/- 3%. Based on this confidence interval, can we state that we are 95% confident that the Senator got a majority of the vote (greater than 50%) and was re-elected? A) Yes B) No Use the information below to answer the questions that follow: The interval is called a confidence interval and the spread of the interval is dependent in part on the spread of the sample statistic around the true population parameter. We know how sample means vary around the true population mean from last lesson when we introduced the sampling distribution of a sample mean. The spread of our sampling distribution depended on the population standard deviation value and the sample size (i.e. the standard deviation of the sampling distribution was derived using a formula that used the population standard deviation and sample size). However, now we do not know the population value for our calculation. We substitute the sample standard deviation value into our formula for the standard deviation of the sampling distribution and now call this measure of spread the standard error of the sample mean. Standard Error of the Sample Mean (SEM) = estimated standard deviation of the sample mean sampling distribution = sample standard deviation/n 9. A random sample of 81 residents of a Central Pennsylvania county found that the average restaurant expenditure in the last month was $60 with a sample standard deviation of $18. The standard error (SE) for the average amount spent will be: A) $18 B) $0.22 C) $2 D) $6.67 10. A random sample of 100 statistics students finds that the average time spent on an exam is 50 minutes with a sample standard deviation of 5 minutes. The standard error (SE) for the average time spent is: A) 10.0 minutes B) 1.0 minutes C) 5.0 minutes D) 0.5 minutes 11. A random sample of 400 statistics students finds that the average time spent on an exam is 50 minutes with a sample standard deviation of 5 minutes. The standard error (SE) for the average time spent is: A) 0.5 minutes B) 0.25 minutes C) 0.0125 minutes D) 5.0 minutes 12. Everything else being equal, an increase in sample size from 100 to 400 will result in the standard error (SE) for the average _________. A) decreasing to of what it was for the sample size of 100 B) decreasing to of what it was for the sample size of 100 C) increasing to 2 times what it was for the sample size of 100 D) increasing to 4 times what it was for the sample size of 100 Use the information below to answer the questions that follow: The confidence interval for our true population mean (for large random samples, n > 30) is given by: sample mean +/- z*SEM Where z* is a multiplier of the SEM that depends on the level of confidence (see Table 10.1 in the online notes for z values for a particular level of confidence). 13. A random sample of 100 full time graduate students in the College of Science is taken to estimate the amount spent on textbooks this semester. The sampled students spent an average of $400 with a sample standard deviation of $100. A 95% confidence interval for the average amount spent by all full time graduate students in the College of Science would be: A) $400 +/- $100 B) $400 +/- $200 C) $400 +/- $10 D) $400 +/- $20 14. A random sample of 225 Penn State football season ticket holders is taken to estimate the population average amount spent in State College on a typical home football game weekend. The sample average was $500 with a sample standard deviation of $75. A 95% confidence interval for the average amount spent by all Penn State Football season ticket holders on a typical home football game weekend is: A) $500 +/- $75 B) $500 +/- $5 C) $500 +/- $10 D) $500 +/- $150 15. A random sample of 81 Statistics undergraduate students found that the average amount of hours of sleep per week during the past semester was 45.5 hours with a sample standard deviation of 3 hours. A 95% confidence interval for the average amount of hours of sleep per week by all Statistics undergraduate students is: A) 45.5 +/- 0.666 hours B) 45.5 +/- 0.333 hours C) 45.5 +/- 3 hours D) 45.5 +/- 6 hours 16. A 95% confidence interval for the average amount spent on textbooks by undergraduate students at a large state university per semester is $400 +/- $50. Would it be correct to state that the true population average could be $360? A) Yes B) No We can also create a confidence interval for the difference between two population proportions or population means. If we were to randomly choose a sample from two different groups (independent samples) and then compare those two sample values to one another and then do this many times with different samples from each group, we would end up with a sampling distribution of the DIFFERENCE between two sample values. This sampling distribution has a measure of spread that is estimated by: standard error of the difference between two sample statistics (SED): = ((standard error of first sample)^2 + (standard error of the second sample)^2) We do not use the SED formula when we have matched pairs (dependent samples). See example 10.7 in the online notes. 17. A consumer testing organization is comparing the battery life of two different models of PCs. They randomly sample 49 of each model of PC (independent samples). The 49 PCs in group 1 have an average battery life of 11 hours with a sample standard deviation of 0.7 hours. The 49 PCs in group 2 have an average battery life of 9 hours with a sample standard deviation of 1.4 hours The standard error for the difference in the two averages is: A) 0.7 + 1.4 = 2.1 hours B) (0.7^2 + 1.4^2) = 1.56 hours C) (0.1^2 + 0.2^2) = 0.223 hours D) 0.1 + 0.2 = 0.3 hours The confidence interval for our true population difference is given by: sample difference +/- z*SED Where z* is a multiplier of the SED that depends on the level of confidence (see Table 10.1 in the online notes for z values for a particular level of confidence). When making conclusions about our population values using a confidence interval of the difference, we look to see if there is a '0' within the interval. If there is a '0' within the interval, we cannot conclude that there is a difference between the two population values. 18. A consumer testing organization is comparing the battery life of two different models of PCs. They randomly sample 49 of each model of PC. The 49 PCs in group 1 have an average battery life of 11 hours with a sample standard deviation of 0.7 hours. The 49 PCs in group 2 have an average battery life of 9 hours with a sample standard deviation of 1.4 hours. The 95% confidence interval of the difference between the two group population means is: A) (11-9) +/- 2 * 0.223 = 2 +/- 0.446 hours B) (11-9) +/- 1 * 0.223 = 2 +/- 0.223 hours C) (11-9) +/- 2 * 1.56 = 2 +/- 3.12 hours D) (11-9) +/- 3 * 0.223 = 2 +/- 0.669 hours 19. As stated in the online notes, holding everything else equal, a four fold increase in sample size alone (as when the sample size increases from 100 to 400) will cut the margin of error in half. Remembering that the margin of error = z*SE (the margin of error is the distance on either side of the sample value that our confidence interval covers), what is the reason that the margin of error is cut in half? 20. What happens to the width of a confidence interval when the confidence level is increased from 90% to 95% (holding all else constant)? Why? (Provide the reasoning behind your answer.) 21. We randomly select 1000 adults from a population of 2 million and also randomly select 1000 adults from a population of 20 million. If the sample standard deviations are the same, how will the margins of error compare for 95% confidence intervals for the true population average? Will the margin of error for the sample from the larger population be greater than, the same, or less than the margin of error for the sample from the smaller population? Why? (Provide the reasoning behind your answer.) 22. We have calculated a 95% confidence interval for the difference between two population proportions. The interval is 4% +/- 5%. Can we conclude that there is a difference between the two group population proportions? Yes or No? Why? (Provide the reasoning behind your answer.) 23. A group of college age males and a group of college age females were sampled. and 95% confidence intervals were created for the true population average of the amount spent on pizza and related items in the past month. The male confidence interval is from $40 to $60 ($50 +/$10). The female confidence interval is from $25 to $35 ($30 +/- $5). Is there a difference between the two group population mean amounts spent? Why? Provide the reasoning behind your answer. (HINT: See page 10.4 in the online notes: "Statistical Significance and Confidence Intervals.")