Ch20-TwoSampleInf

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Practice Problems – Chapter 20 Two-Sample Inference Chapter 20 Exercises, #3 Canada, deux Suppose an advocacy organization surveys 960 Canadians and 192 of them reported being born in another country (www.unitednorthamerica.org/simdiff.htm). Similarly, 170 out of 1250 U.S. citizens reported being foreign-born. This data was used to create a 95% two-proportion confidence interval for the difference between Canadians and U.S. citizens who were born in foreign countries (provided below). Interpret this interval with a sentence in context. 95% ????𝑖????? 𝑖??????𝑙 ??? ? 𝐶𝑎?𝑎?𝑖𝑎?? − ? 𝐴???𝑖?𝑎?? 𝑖? (3.24%, 9.56%) Chapter 20 Exercises, #11, 13 Cost of shopping Do consumers spend more on a trip to Walmart or Target? Suppose researchers interested in this question collected a systematic sample from 85 Walmart customers and 80 Target customers by asking customers for their purchase amount as they left the stores. The data collected are summarized in the table below. Walmart Target n 85 80 ? ̅ $45 $53 s $21 $19 a) To perform inference on these two samples, what conditions must be met? Are they? Explain. b) Researchers calculated a 95% confidence interval for the mean difference between Walmart and Target purchase amounts. The interval was (−$14.15, $1.85). Explain in context what this interval means. Chapter 20 Exercises, #21 Online social networking In September 2013, the Pew Internet and American Life Project surveyed American adults on their Facebook use. It found that 64% visited the site on a daily basis, up from 51% in 2010. What does it mean to say that the difference in proportions is “significant”? Chapter 20 Exercises, #23 Revealing information Eight hundred eighty-six randomly sampled teens were asked which of several personal items of information they thought it ok to share with someone they had just met. Forty-four percent said it was ok to share their e-mail addresses, but only 29% said they would give out their cell phone numbers. A researcher claims that a two-proportion z-test could tell whether there was a real difference among all teens. Explain why that test would not be appropriate for these data. Chapter 20 Exercises, #27 Arthritis The Centers for Disease Control and Prevention reported a survey of randomly selected Americans age 65 and older, which found that 411 of 1012 men and 535 of 1062 women suffered from some form of arthritis.
a) Are the assumptions and conditions necessary for inference satisfied? Explain. b) A 95% confidence interval for the difference in the proportions of senior men and women who have this disease is found to be (0.055, 0.140) (for practice, feel free to check this!) . Interpret this interval in this context. c) Does this confidence interval suggest that arthritis is more likely to afflict women than men? Explain. Chapter 20 Exercises, #41 Mammograms It’s widely believed that regular mammogram screening may detect breast cancer early, resulting in fewer deaths from that disease. One study that investigated this issue over a period of 18 years was published during the 1970s. Among 30,565 women who had never had mammograms, 196 died of breast cancer, while only 153 of 30,131 who had undergone screening died of breast cancer. a) Do these results suggest that mammograms may be an effective screening tool to reduce breast cancer deaths? Assume 𝛼 = 0.05 and the standardized difference of the proportions ? ????? ? ???????? is 2.17. b) If your conclusion is incorrect, what type of error have you committed? What would this mean? Chapter 20 Exercises, #47 Sensitive men An article in Time magazine, reporting on a survey of men’s attitudes, noted that “Young men are more comfortable than older men talking about their problems.” The survey reported that 80 of 129 surveyed 18- to 24-year-old men and 98 of 184 25- to 34-year-old men said they were comfor table. What do you think? Is Time’s interpretation justified by these numbers? Assume 𝛼 = 0.05 and the standardized difference of the proportions ? ????𝑔 − ? ??? is 1.56. Chapter 20 Exercises, #53 Hot dogs, second helping In the July 2007 issue, Consumer Reports examined the fat content (in grams) of two kinds of hot dogs: meat (usually a mixture of pork, turkey, and chicken) and all beef. The researchers purchased samples of several different brands. The resulting 90% confidence interval for μ Meat −μ Beef is (−6.5,−1.4). a) The endpoints of this confidence interval are negative numbers. What does that indicate? b) What does the fact that the confidence interval does not contain 0 indicate? Chapter 20 Exercises, #55 Hot dogs, last one In Exercise 53, we saw a 90% confidence interval of (−6.5,−1.4) grams for μ Meat −μ Beef the difference in mean fat content for meat vs. all-beef hot dogs. Explain why you think each of the following statements is true or false: a) If I eat a meat hot dog instead of a beef dog, there’s a 90% chance I’ll consume less fat. b) 90% of meat hot dogs have between 1.4 and 6.5 grams less fat than a beef hot dog. c) I’m 90% confident that meat hot dogs average between 1.4 and 6.5 grams less fat than the beef hot dogs.
d) If I were to get more samples of both kinds of hot dogs, 90% of the time the meat hot dogs would average between 1.4 and 6.5 grams less fat than the beef hot dogs. e) If I tested more samples, I’d expect about 90% of the resulting confidence intervals to include the true difference in mean fat content between the two kinds of hot dogs. Chapter 20 Exercises, #83 Tees Does it matter what kind of tee a golfer places the ball on? The company that manufactures “Stinger” tees claims that the thinner shaft and smaller head lessen resistance and drag, reducing spin and allowing the ball to travel farther. In August 2003, Golf Laboratories, Inc., compared the distance traveled by golf balls hit off regular wooden tees to those hit off Stinger tees. All the balls were struck by the same golf club using a robotic device set to swing the club head at approximately 95 miles per hour. Summary statistics from the test are shown in the table below. Assume that 6 balls were hit off each tee and that the data were suitable for inference. Is there evidence that balls hit off the Stinger tees have a higher initial velocity? Assume 𝛼 = 0.05 and the standardized difference of the means 𝜇 ??𝑔??𝑎? 𝜇 ??𝑖?𝑔?? is -4.57. Blank Blank Total Distance (yards) Ball Velocity (mph) Club Velocity (mph) Regular Tee Avg. 227.17 127.00 96.17 Blank SD 2.14 0.89 0.41 Stinger Tee Avg. 241.00 128.83 96.17 Blank SD 2.76 0.41 0.52 Chapter 20 Exercises, #85 Music and memory Is it a good idea to listen to music when studying for a big test? In a study conducted by some statistics students, 62 people were randomly assigned to listen to rap music, music by Mozart, or no music while attempting to memorize objects pictured on a page. They were then asked to list all the objects they could remember. Here are summary statistics for each group: Blank Rap Mozart No Music Count 29 20 13 Mean 10.72 10.00 12.77 SD 3.99 3.19 4.73 Does it appear that it is better to study while listening to Mozart than to rap music? Test an appropriate hypothesis and state your conclusion. Assume 𝛼 = 0.05 , the conditions are satisfied, and the standardized difference of the means 𝜇 𝑀??𝑎?? − 𝜇 𝑅𝑎? is -0.70.
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Other Questions: Use the following information to answer the next 6 exercises : Indicate if the hypothesis test is for A. two means (independent samples) B. matched or paired samples C. two proportions D. single mean E. single proportion 1. Sample statistics suggest that 70% of males pass their drivers test in the first attempt, while 65% of females pass the test in the first attempt. Of interest is whether the proportions are, in fact, equal. 2. A new laundry detergent is tested on consumers. Of interest is the proportion of consumers who prefer the new brand over the leading competitor. A study is done to test this. 3. A new windshield treatment claims to repel water more effectively than without treatment. Ten windshields are tested by simulating rain without the new treatment. The same windshields are then treated, and the experiment is run again. A hypothesis test is conducted. 4. According to a television commercial, 80% of dentists agree that Ultrafresh toothpaste is the best on the market. Test this claim. 5. A new medicine is said to help improve sleep. Eight subjects are picked at random and given the medicine. The mean hours slept per night for each person over a 4-week period were recorded before starting the medication and again (another 4 weeks) after. 6. It is thought that teenagers sleep more than adults on average. A study is done to verify this. A sample of 16 teenagers has a mean of 8.9 hours slept and a standard deviation of 1.2. A sample of 12 adults has a mean of 6.9 hours slept and a standard deviation of 0.6.
7. Consider an experiment involving the comparison of the mean heart rate following 30 minutes of aerobic exercise among females aged 20 to 24 years as compared to females aged 30-34 years. A 95% confidence interval for the difference in the mean heart rates of females age 20-24 and females age 30- 34 is found to be (-13.776, 32.016) beats per minute (bpm). Fill in the blanks in the statement below to answer the question: Do the mean heart rates for females in these age groups differ? We would conclude that the mean heart rate of age 20-24 (does / does not) differ because the 95% CI for 𝜇 1 − 𝜇 2 _________________________________. 8. A method of analyzing digital music files to determine the key in which the music was written was tested to determine its accuracy. In a sample of 307 pop music selections, the key was correctly identified in 245 of them. In a sample of 347 new age sections, the key was correctly identified in 304 of them. A 99% confidence interval for the difference in the proportions of correctly identified keys in pop and new age music is found to be −0.0781 ± 0.0745 = (−0.1526, −0.0036) . Can we conclude that these proportions are significantly different? Explain.
Key Chapter 20 Exercises, #3 We are 95% confident that, based on these data, the proportion of foreign-born Canadians is between 3.24% and 9.56% higher than the proportion of foreign-born Americans. Chapter 20 Exercises, #11, 13 a) We must assume the samples were random or otherwise independent of each other. We also assume that the distributions are roughly Normal, so it would be a good idea to check a histogram to make sure there isn’t strong skewness or outliers. b) We are 95% confident that the mean purchase amount at Walmart is between $1.85 and $14.15 less than the mean purchase amount at Target. Chapter 20 Exercises, #21 T o say that the difference in proportions is “significant” means: It’s very unlikely that samples would show an observed difference this large if in fact there is no real difference in the proportions of American adults using Facebook on a daily basis between the two time periods. Chapter 20 Exercises, #23 The responses are not from two independent groups, but are from the same individuals. Chapter 20 Exercises, #27 a) Yes. Random samples; less than 10% of the population; samples are independent; more than 10 successes and failures in each sample. b) We are 95% confident that the proportion of American women age 65 and older who suffer from arthritis is between 5.5% and 14.0% higher than the proportion of American men of the same age who suffer from arthritis. c) Yes. The entire interval lies above 0. Chapter 20 Exercises, #41 a) 𝐻 0 : ? ????? − ? ???????? ≤ 0, ?? 𝐻 𝐴 : ? ????? − ? ???????? > 0 ? = 2.17 ? − 𝑉?𝑙?? = 𝑃(? > 2.17) = 0.0148 With a P-value less than alpha, we reject 𝐻 0 . There is enough evidence at the 5% significance level to suggest that mammograms likely do reduce breast cancer deaths (i.e. the proportion of women who never had mammograms who die of breast cancer is greater than the proportion of women who undergo screening who die of breast cancer). b) Type I Error, since the null hypothesis was rejected. What this error would mean: We conclude that mammograms help reduce breast deaths, when actually they do not.
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Chapter 20 Exercises, #47 𝐻 0 : ? ????𝑔 − ? ??? ≤ 0, ?? 𝐻 𝐴 : ? ????𝑔 − ? ??? > 0 ? = 1.56 ? − 𝑉?𝑙?? = 𝑃(? > 1.56) = 0.0602 With a P-value greater than alpha, we fail to reject 𝐻 0 . There is not enough evidence at the 5% significance level to support Time’s statement that y oung men are more comfortable than older men talking about their problems (That is, we do not support the statement that the proportion of young men who are comfortable is greater than the proportion of older men who are comfortable talking about their problems). Chapter 20 Exercises, #53 a) Plausible values of μ Meat −μ Beef are all negative, so the mean fat content is probably higher for beef hot dogs. b) The difference in sample means is significant. Chapter 20 Exercises, #55 a) False. The confidence interval is about means, not about individual hot dogs. b) False. The confidence interval is about means, not about individual hot dogs. c) True. d) False. CIs based on other samples will also try to estimate the true difference in population means; there’s no reason to expect other samples to conform to this result. e) True. Chapter 20 Exercises, #83 𝐻 0 : 𝜇 ??𝑔??𝑎? − 𝜇 ??𝑖?𝑔?? ≥ 0, ?? 𝐻 𝐴 : 𝜇 ??𝑔??𝑎? − 𝜇 ??𝑖?𝑔?? < 0 ? = −4.57, ?? = 5 ? − 𝑉?𝑙?? = 𝑃(? 5 < −4.57) = 0.0030 With a P-value less than alpha, we reject 𝐻 0 . There is sufficient evidence at the 5% significance level to suggest that golf balls hit off Stinger tees have a higher mean initial velocity. Chapter 20 Exercises, #85 𝐻 0 : 𝜇 𝑀??𝑎?? − 𝜇 𝑅𝑎? ≤ 0, ?? 𝐻 𝐴 : 𝜇 𝑀??𝑎?? − 𝜇 𝑅𝑎? > 0 ? = −0.70, ?? = 19 ? − 𝑉?𝑙?? = 𝑃(? 19 > −0.70) = 0.7538 Because the P-value is so large, we fail to reject 𝐻 0 . These data do not provide evidence that listening to Mozart while studying is better than listening to rap.
***Note for Exercise 85: Now you might have noticed that subjects listening to Mozart recalled fewer items. And so, there is no need for a statistical test. Looking and thinking first can save you some calculation effort. If you did notice, and said there’s no evidence of a benefi t, without calculation —that’s a good answer to this exercise. Other Questions: 1. Two proportions. 2. Single proportion. 3. Matched or paired samples (the same windshields are tested). 4. Single proportion. 5. Matched or paired samples (the same subjects are measured twice, before and after). 6. Two means (independent). 7. We would conclude that the mean heart rate of age 20-24 (does / does not) differ because the 95% CI for 𝜇 1 − 𝜇 2 ______ contains the value 0 ________. *Note: Since 0 is within the interval limits, it is a plausible value for the difference. If 𝜇 1 − 𝜇 2 = 0 is plausible, then it is plausible that 𝜇 1 = 𝜇 2 . 8. Since 0 is not in this interval, we conclude that these proportions are significantly different. Plausible values for the difference do not include 0, meaning they are not equal. If ? 1 − ? 2 ≠ 0 , then ? 1 ≠ ? 2 .