Lesson 10 Question 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: 4.0% 4.9% 0.24% 24% Question 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: 4.9% 4.0% 2.45% 24% Question 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: as large as it was before the sample size increase 2 times larger than it was before the sample size increase 4 times larger than it was before the sample size increase as large as it was before the sample size increase Question 4 As sample size increases: the SE increases the SE remains the same the SE decreases Question 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: 40% +/- 2.45% 40% +/- 4.90% 95% +/- 40% 5% +/- 40% Question 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: 51% +/- 5% 51% +/- 2.5% 95% +/- 5% 95% +/- 2.5% Question 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: 80% +/- 5.33% 80% +/- 2.666% 80% +/- 8% 80% +/- 2% Question 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? Yes No Question 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: $18 $0.22 $2 $6.67 Question 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: 10.0 minutes 1.0 minutes 5.0 minutes 0.5 minutes Question 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: 0.5 minutes 0.25 minutes 0.0125 minutes 5.0 minutes Question 12 Everything else being equal, an increase in sample size from 100 to 400 will result in the standard error (SE) for the average _________. decreasing to of what it was for the sample size of 100 decreasing to of what it was for the sample size of 100 increasing to 2 times what it was for the sample size of 100 increasing to 4 times what it was for the sample size of 100 Question 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: $400 +/- $100 $400 +/- $200 $400 +/- $10 $400 +/- $20 Question 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: $500 +/- $75 $500 +/- $5 $500 +/- $10 $500 +/- $150 Question 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: 45.5 +/- 0.666 hours 45.5 +/- 0.333 hours 45.5 +/- 3 hours 45.5 +/- 6 hours Question 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? Yes No Question 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: 0.7 + 1.4 = 2.1 hours (0.7^2 + 1.4^2) = 1.56 hours (0.1^2 + 0.2^2) = 0.223 hours 0.1 + 0.2 = 0.3 hours Question 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: (11-9) +/- 2 * 0.223 = 2 +/- 0.446 hours (11-9) +/- 1 * 0.223 = 2 +/- 0.223 hours (11-9) +/- 2 * 1.56 = 2 +/- 3.12 hours (11-9) +/- 3 * 0.223 = 2 +/- 0.669 hours Question 19 For large random samples, when we look at sample means or proportions ... What happens to the width of a confidence interval when the confidence level is increased from 68% to 95% (holding all else constant)? Explain briefly. Question 20 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? Explain briefly. Question 21 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 significant difference between the two group population proportions? Yes or No? Explain briefly. Question 22 A random sample of 900 homeowners in Allegheny County is taken in order to estimate the average amount of money spent by homeowners on landscaping this spring. It turns out that this group spent an average of $200 with a standard deviation of $150. A 95% confidence interval for the average amount spent on landscaping by all Allegheny County homeowners is calculated to be $200 $10. Which is true about this situation? The true average amount spent must be in this interval. The procedure used here produces intervals that cover the right answer for 95% of all samples. This confidence interval won't cover the right answer since a histogram of the amount spent is skewed. The interval would be wider if we used an 80% confidence level instead of 95%. Question 23 A random sample of 900 homeowners in Allegheny County is taken in order to estimate the average amount of money spent by homeowners on landscaping this spring. It turns out that this group spent an average of $200 with a standard deviation of $150. In this situation, what is the population? What is the parameter? What is the sample? What is the statistic? Question 24 One thousand students are randomly selected from the list of those currently registered at Harvard University, and they each report how many miles they rode in an MTA bus during the past week. Because many students did not ride a bus at all, a histogram of the values does not look like the normal curve. True or False: Even though the histogram did not follow the normal curve, it is still possible to use the normal curve to make a confidence interval for the average number of miles that all Harvard students rode on MTA buses last week. (explain briefly why you chose your answer). Use for Questions 1-6: A random sample of 100 males found that 80% watched a particular TV show in the past week. A random sample of 100 females found that 50% watched the same TV show in the past week Question 1 The standard error for the sampling distribution of the sample proportion for males is: 0.04 0.0016 0.08 0.16 Question 2 The standard error for the sampling distribution of the sample proportion for females is: 0.05 0.25 0.0025 0.10 Question 3 What is the standard error for the difference between our two sample values? ((0.04)2 + (0.05)2) = 0.0041 = 0.064 (0.04 + .05) = 0.09 = 0.30 ((0.0016)2 + (0.0025)2) = 0.000009 = 0.003 (0.08 + 0.10) = 0.18 = .42 Question 4 What is the formula for a 95% confidence interval of the difference between the two population proportions? (sample proportion 1 - sample proportion 2) +/- 2 *SED (sample proportion 1 - sample proportion 2) +/- 1*SED (sample proportion 1 - sample proportion 2) +/- 3*SED (sample proportion 1 - sample proportion 2) +/- 2 Question 5 What is the 95% confidence interval of the difference between the two population proportions? (.80 - .50) +/- 2 * 0.064 = .30 +/- 0.128 = (0.172 to 0.428) = (17.2% to 42.8%) (.80 - .50) +/- 1 * 0.003 = .30 +/- 0.003 = (0.297 to 0.303) = (29.7% to 30.3%) (.80 - .50) +/- 2 * 0.003 = .30 +/- 0.006 = (0.294 to 0.306) = (29.4% to 30.6%) (.80 - .50) +/- 3 * 0.003 = .30 +/- 0.009 = (0.291 to 0.309) = (29.1% to 30.9%) (.80 - .50) +/- 1 * 0.064 = .30 +/- 0.064 = (0.236 to 0.364) = (23.6% to 36.4%) (.80 - .50) +/- 3 *0.064 = .30 +/- 0.192 = (0.108 to 0.492) = (10.8% to 49.2%) Question 6 The confidence interval in #5 shows that: the difference between the population proportions for the two groups is significant the difference between the population proportions for the two groups is not significant cannot tell from the information given Question 7 A random sample of 100 full time students at a large two-year college is taken to estimate the amount they spent on textbooks this semester. The sampled students spent an average of $300 with a standard deviation of $50. Thus, a 95% confidence interval for the average amount spent by all full time students at that school would be: $300 +/- $0.50 $300 +/- $5.00 $300 +/- $10.00 $300 +/- $50.00 Question 8 A random sample of 225 male students at a large two-year college is taken to estimate the height of male students at this school. The sampled male students have an average height of 68 inches with a standard deviation of 9 inches. A random sample of 144 female students is also taken from the same school to estimate the height of female students at this school. The sampled female students have an average height of 64 inches with a standard deviation of 3 inches. A 95% confidence interval for the difference in average height between males and females is given by: 4 +/- 0.92195 inches 4 +/- 1.30 4 +/- 1.844 inches 4 +/- 0.65 inches Lesson 11 Question 1 A polling group surveyed a city in Scotland regarding residents' opinions on independence from the UK. The poll wants to find out whether there is strong evidence to show that a majority (> 50%) of the residents will vote 'Yes.' The survey polled 2000 residents, of which 1050 responded that they will vote 'Yes' on Scotland independence (52.5%). What are the null and alternative hypotheses? Null: the percentage of 'Yes' votes is 52.5%; Alternative: the percentage of 'Yes' votes is greater than 52.5% Null: the percentage of 'Yes' votes is greater than 52.5%; Alternative: the percentage of 'Yes' votes is 52.5% Null: the percentage of 'Yes' votes is 50%; Alternative: the percentage of 'Yes' votes is greater than 50% Null: the percentage of 'Yes' votes is greater than 50%; Alternative: the percentage of 'Yes' votes is 50% Question 2 For patients with a particular disease, the population proportion of those successfully treated with a standard treatment that has been used for many years is 0.75. A medical research group invents a new treatment that they believe will be more successful, i.e. the population proportion will exceed 0.75. A doctor plans a clinical trial he hopes will prove this claim. A sample of 100 patients with the disease is obtained. Each person is treated with the new treatment and eventually classified as having either been successfully or not successfully treated with the new treatment. Out of 100 patients, 80 (80%) were successfully treated by the new treatment. What are the null and alternative hypotheses? Null: the population proportion of those successfully treated by the new treatment exceeds 0.75 (p > 0.75); Alternative: the population proportion of those successfully treated by the new treatment is 0.75 (p = 0.75) Null: the population proportion of those successfully treated by the new treatment is 0.75 (p = 0.75); Alternative: the population proportion of those successfully treated by the new treatment exceeds 0.75 (p > 0.75) Null: the population proportion of those successfully treated by the new treatment is 0.80 (p = 0.80); Alternative: the population proportion of those successfully treated by the new treatment exceeds 0.80 (p > 0.80) Null: the population proportion of those successfully treated by the new treatment exceeds 0.80 (p > 0.80); Alternative: the population proportion of those successfully treated by the new treatment is 0.80 (p = 0.80) Question 3 Suppose that a study is done comparing two different contact lens wetting solutions with regard to hours of wearing comfort. 100 contact lens wearers are randomly divided into two groups. One group uses solution A for 2 months. The other group uses solution B for 2 months. The researcher wants to determine if there is a difference in the hours of wearing comfort for the two groups. The population mean number of hours of wearing comfort will be compared for the two groups. What are the null and alternative hypotheses being tested by the researcher? Null: there is a difference in the population mean number of hours of wearing comfort for the two groups (two population means are not equal); Alternative: there is no difference in the population mean number of hours of wearing comfort for the two groups (two population means are equal). Null: there is no difference in the population mean number of hours of wearing comfort for the two groups (two population means are equal); Alternative: the population mean from group A is larger than that of group B (population mean group A > population mean group B) Null: there is no difference in the population mean number of hours of wearing comfort for the two groups (two population means are equal); Alternative: there is a difference in the population mean number of hours of wearing comfort for the two groups (two population means are not equal) Null: the population mean number of hours of wearing comfort for group A is less than that of group B (population mean A population mean A) Question 4 A car company is testing to see if the proportion of all adults who prefer blue cars has changed (differs) from 0.35 since industry statistics indicate this proportion has been 0.35 for quite some time. A random sample of 1,000 car owners finds that the proportion that prefers blue cars is 0.40. What are the null and alternative hypotheses being tested? Null: the population proportion who prefer blue cars is 0.35; Alternative: the population proportion who prefer blue cars is greater than 0.35 Null: the population proportion who prefer blue cars is 0.40; Alternative: the population proportion who prefer blue cars differs from (does not equal) 0.40 Null: the population proportion who prefer blue cars is 0.40; Alternative: the population proportion who prefer blue cars is greater than 0.40 Null: the population proportion who prefer blue cars is 0.35; Alternative: the population proportion who prefer blue cars differs from (does not equal) 0.35 Question 5 A college admissions website tells potential students that the average amount spent on textbooks is $300.00 per semester. A student group decides to research book cost because it now believes that book cost per semester is greater than $300.00. A random sample of 225 current students finds that the sample average is $324 with a standard deviation of $60. In this situation, the null hypothesis is that: the average for all students is greater than $300 the average for all students is greater than $324 the average for all students equals $300 the average for all students equals $324 Question 6 A college admissions website tells potential students that the average amount spent on textbooks is $300.00 per semester. A student group decides to research book cost because it now believes that book cost per semester is greater than $300.00. A random sample of 225 current students finds that the sample average is $324 with a standard deviation of $60. What is the standard error of the mean under the null hypothesis? $60 $4 $8 $15 Question 7 A college admissions website tells potential students that the average amount spent on textbooks is $300.00 per semester. A student group decides to research book cost because it now believes that book cost per semester is greater than $300.00. A random sample of 225 current students finds that the sample average is $324 with a standard deviation of $60. What is our test statistic value (standardized score)? +24 +6 -6 0.4 Question 8 A polling organization surveyed Pennsylvanians regarding residents' opinions on whether there should be term limits for State Representatives and Senators. The polling organization wants to find out if there is strong evidence that a majority (> 50%) of the residents are in favor of term limits. The pollsters randomly polled 500 residents, of which 275 responded that they favor term limits (55%). What approximately is the standard error of the proportion (SEP) under the null hypothesis? 0.044 or 4.4% 0.011 or 1.1% 0.022 or 2.2% 0.05 or 5% Question 9 A car company is testing to see if the proportion of all adults who prefer blue cars has changed (differs) from 0.35 since industry statistics indicate this proportion has been 0.35 for quite some time. A random sample of 1,000 car owners finds that the proportion that prefers blue cars is 0.40. What is our test statistic value (standardized score)? 0.40 0.05 -3.33 +3.33 Question 10 A consultant believes that the average amount spent by customers at an online shoe store will be less than the current $100 if shipping costs are increased by 10% (the projected shipping cost increase proposed by their carrier). To test the null hypothesis that the population mean = $100 versus the alternative hypothesis that the population mean < $100, the consultant conducts a study using a large random sample and calculates a test statistic of z = +0.5. The p-value for this test would be 0.05 0.84 0.16 0.31 Question 11 A consultant believes that the average amount spent by customers at an online book store will be more than the current $100 if they add a new faster shipping service. To test the null hypothesis that the population mean = $100 versus the alternative hypothesis that the population mean > $100, the consultant conducts a study using a large random sample and calculates a test statistic of z = +1.96. The p-value for this test would be which of the following? 0.975 0.05 0.025 0.95 Question 12 A cereal manufacturer tests their equipment weekly to be assured that the proper amount of cereal is in each box of cereal. The company wants to see if the amount differs from the stated amount on the box. The stated amount on each box for this particular cereal is 12.5 ounces. The manufacturer takes a random sample of 100 boxes and finds that they average 12.2 ounces with a standard deviation of 3 ounces. Is this a one-tailed or two-tailed significance testing situation? One-tailed with an alternative that the machine fills and average greater than 12.5 ounces One-tailed with an alternative that the machine fills and average less than 12.5 ounces Two-tailed with an alternative that the machine fills and average different from 12.5 ounces Whether it is one-tailed or two-tailed will depend on the p-value Question 13 A cereal manufacturer tests their equipment weekly to be assured that the proper amount of cereal is in each box of cereal. The company wants to see if the amount differs from the stated amount on the box. The stated amount on each box for this particular cereal is 12.5 ounces. The manufacturer takes a random sample of 100 boxes and finds that they average 12.2 ounces with a standard deviation of 3 ounces. The pvalue is then about 3% 16% 32% 62% Question 14 Suppose that the null hypothesis is, \"The population mean is $200,\" and the alternative hypothesis is, \"The population mean is less than $200.\" Also, suppose the test statistic value is -0.50, with a p-value of 0.31. The p-value is the probability of obtaining our test statistic value or a value even smaller, if in fact the population mean is less than $200 obtaining our test statistic value or a value even larger, if in fact the population mean is less than $200 obtaining our test statistic value or a value even smaller, if in fact the population mean is $200 obtaining our test statistic value or a value even larger, if in fact the population mean is $200 Question 15 Suppose that the null hypothesis is, \"The population proportion is 0.50,\" and the alternative hypothesis is, \"The population proportion is greater than 0.50.\" Further, suppose that our test statistic is +1.96, with a p-value of 0.025. The p-value is the probability of obtaining our test statistic value of 1.96 or larger, if in fact the population proportion is 0.50 obtaining our test statistic value of 1.96 or smaller, if in fact the population proportion is 0.50 obtaining our test statistic value of 1.96 or larger, if in fact the population proportion is greater than 0.50 obtaining our test statistic value of 1.96 or smaller, if in fact the population proportion is greater than 0.50 Question 16 A p-value of 0.05 or less is said to indicate that the results are \"statistically significant.\" What does statistically significant mean? The null hypothesis is a poor explanation of the data. The null hypothesis is a good explanation of the data. The alternative hypothesis is a poor explanation of the data. Question 17 Three of the following statements about a p-value are true. Which one is false? If we got a p-value of 0.52, we would not reject the null hypothesis. If the p-value is very small, we reject the null hypothesis. The p-value is the probability that the null hypothesis is true. The p-value is the probability, assuming the null hypothesis is true, of seeing results as (or more) extreme as what we observed in the sample. Question 18 Once you reject the null hypothesis, is it possible to have a Type 2 error? No Yes Question 19 A researcher conducts a randomized experiment to test whether a diet that includes 100 grams of blueberries each day will improve the function of the cathelicidin antimicrobial peptide gene which is known to have an important role in the immune system. The gene's function is measured in 40 subjects who added 100 grams of blueberries to their daily diet and 40 control subjects who ate similar meals but without the blueberries. At the end of the experiment, a significance test is conducted and the p-value is found to be 0.0042. This says that: Assuming the gene's function is the same regardless of whether you eat blueberries daily is a poor explanation of the data. Assuming the gene's function is the same regardless of whether you eat blueberries daily is a good explanation of the data. Eating blueberries has been proven to help your immune system. Eating blueberries has been proven to be unimportant in improving your immune system. Question 20 If the null hypothesis is true then it can not be rejected. True False Question 21 What are the null and alternative hypotheses in this situation? Question 22 What is the standard deviation of the proportion of times the older child scores higher if the null hypothesis is true? Explain how you got your answer. Question 23 Is the normal approximation appropriate to use in this case? Explain briefly. Question 24 What is the test statistic and the calculated p-value? Explain how you got your answer. Question 25 What is the conclusion you should draw from these results? Questions 1-7: Previously a study found that Statistics students were spending about $400.00 on textbooks per semester. A researcher at Penn State decides to research book cost because he now believes that book cost per semester is greater than $400.00. A random sample of 64 Statistics students finds that the sample average is $410.00 with a standard deviation of $80. Is this strong evidence that the average amount spent on textbooks per semester is greater than $400? We wish to carry out the appropriate hypothesis test. Question 1 In this situation, the null hypothesis is that: the long run average equals $400 the long run average is greater than $400 the long run average is greater than $410 the long run average equals $410 Question 2 The standard error of the mean under the null hypothesis is: $10 $80 $20 $15 Question 3 What is our test statistic value (standardized score)? +1.0 +0.5 -1.0 -0.5 Question 4 What is the p-value of our test statistic (standardized score) for the significance test? 0.16 or 16% 0.31 or 31% 0.84 or 84% 0.69 or 69% Question 5 Based on the p-value, we can conclude: that the null hypothesis is a reasonable explanation of the data that the null hypothesis is a poor explanation of the data that the null hypothesis is true that the alternative hypothesis is true Question 6 What is the Type 2 error in this situation? fail to reject that the population average is $400 when we should reject that the population average is $400 reject that the population average is $400 when we should not reject that the population average is $400 Question 7 Given the p-value found in our hypothesis test, would a 95% confidence interval for the true population average spent include $400? Yes No Question 8 What would the p-value be if our alternative hypothesis was that textbook costs are \"different than\" $400 instead of \"greater than\" $400? (Note: we consider a hypothesis test where we are testing whether the population value is greater than (>) some value to be a one-sided test. When we test to see if the population value is different (not equal to) some value, then we have a two-sided test) the same as the p-value we found for the one-sided test less than the p-value we found for the one-sided test less than the p-value we found for the one-sided test greater than the p-value we found for the one-sided test cannot tell until we calculate the p-value Question 9 If the sample size was increased and instead of randomly selecting 64 students, 225 students were randomly selected. The p-value of our test statistic (standardized score) would___ be larger than before be smaller than before be the same as before cannot tell from the information given