What is a distribution of means (from a set of samples)?
> a. Within a group of 500 typical first-year college students, approximately how many have loans to help pay for their education? b. Can you conclude that some students who are satisfied with their coursework also text in class? Explain. Figure 3.19
> Sample size, n = 100; sample mean, x̅ = 55.0 cm; sample standard deviation, s = 5.0 cm
> The National Health Examination involves measurements from about 25,000 people, and the results are used to estimate values of various population means. Is it valid to criticize this survey because the sample size is only about 0.01% of the population of
> Here is a typical statement made by the media: “Based on a recent study, pennies weigh an average of 2.5 grams with a margin of error of 0.006 gram.” What important and relevant piece of information is omitted from that statement? Is it acceptable to use
> Based on a random sample of 48 birth weights of girls born at Albany Medical Center Hospital, the sample mean is 2965 grams and the margin of error for a 95% confidence interval is 245 grams. Find the 95% confidence interval.
> One study involved a sample of people’s body temperatures and it resulted in this 95% confidence interval for the mean: 98.1 ⁰F
> Although it made our study more expensive, we chose a larger sample size in order to have a smaller margin of error.
> Based on our sample, the 95% confidence interval for the mean amount of television watched by adult Americans is 2.5 to 2.7 hours per day. Therefore, there is a 95% chance that the mean for all Americans will fall somewhere in this range and a 5% chance
> Based on our sample, the 95% confidence interval for the mean amount of television watched by adult Americans is 2.5 to 2.7 hours per day. Therefore, 95% of all Americans watch between 2.5 and 2.7 hours of TV per day. Answer: The statement does not ma
> An engineering student measures the strength of fingers used to push buttons by testing her own family members.
> Based on our sample, the 95% confidence interval for the mean amount of television watched by adult Americans is 2.5 to 2.7 hours per day. Therefore, there is 95% chance that the actual mean for the population is 2.6 hours.
> a. In about what year did (will) the population of 45- to 54-year-olds peak as a percentage of the population? b. In about what year did (will) the population of over 65-year-olds peak as a percentage of the population? c. Discuss any significant trends
> I selected three different samples of size n = 10 drawn from the 1500 students at my school, and with these I constructed the sampling distribution.
> How does the sample size affect how close to normal a distribution of either sample means or sample proportions will be? What are the means and standard deviations of the distributions in each case?
> What is a sample mean? What is a sample proportion? Summarize the notation used for these statistics.
> What is a sampling error? How does it differ from other sources of error? In general, how does the sampling error increase or decrease with larger sample sizes? Explain.
> Distinguish between a distribution of sample means and a distribution of sample proportions.
> When 650 adults are randomly selected, find the probability that 84 or more of them are left handed. Does the result of 84 left handed adults appear to be unusually high?
> If 820 adults are randomly selected, find the probability that 92 or more of them are left handed. Does the result of 92 left handed adults appear to be unusually high?
> If 250 adults are randomly selected, find the probability that 15 or fewer of them are left handed. Does the result of 15 left handed adults appear to be unusually low?
> If 500 adults are randomly selected, find the probability that 45 or fewer of them are left handed. Does the result of 45 left handed adults appear to be unusually low?
> A medical researcher is conducting a study to test the effectiveness of a drug designed to lower cholesterol levels. She randomly selects a sample of 100 males and 100 females.
> a. About what percentage of the population were over age 65 in 2010? b. About what percentage of the population is projected to be over age 65 in 2050? c. Describe the projected change in the 45–54 age groups between 2010 and 2050. Figu
> A random sample of 81 newborn girls is obtained and they have a mean birth weight of 2919 grams. What is the probability of randomly selecting another 81 newborn girls and getting a mean birth weight that is 2919 grams or less? Does it seem like a sample
> A random sample of 64 newborn girls is obtained and they have a mean birth weight of 3390 grams. What is the probability of randomly selecting another 64 newborn girls and getting a mean birth weight that is 3390 grams or more? Does it seem like a sample
> A random sample of 36 newborn girls is obtained and they have a mean birth weight of 3272 grams. What is the probability of randomly selecting another 36 newborn girls and getting a mean birth weight that is 3272 grams or more? Does it seem like a sampl
> A random sample of 100 newborn girls is obtained and they have a mean birth weight of 2966 grams. What is the probability of randomly selecting another 100 newborn girls and getting a mean birth weight that is 2966 grams or lower? Does it seem like a sa
> The ages (in years) of the four U.S. presidents when they were assassinated in office are 56(Lincoln), 49 (Garfield), 58 (McKinley), and 46 (Kennedy). Consider these four ages to be a population. a. Assuming that two of the ages are randomly selected to
> A quarterback threw 1 interception in his first game, 2 interceptions in his second game, and 5 interceptions in his third game, and then he retired. Consider the values 1, 2, and 5 to be a population. Assume that samples of size 2 are randomly selected
> The College of Portland has a total enrollment of N = 2444 students and 269 of them are left handed. You conduct a survey of n = 50 students and find that 8 of them are left handed. a. What is the population proportion, p, of left handed students? b. Wha
> A population consists of a batch of 25,344 aspirin tablets, and it includes 1,014 that are defective because they do not meet specifications. A random sample of n = 250 of the tablets is obtained and tested, with the result that 18 of them are defective.
> Assume that the population of heights of adult males has a normal distribution with a mean of µ= 174.1 centimeters (cm) and a standard deviation of σ = 7.1 cm. a. If a sample of size n = 100 adult males results in a mean height of x̅ = 172.0 cm, how many
> Assume that cans of cola are filled such that the actual amounts have a population mean of µ = 12.00 ounces. A random sample of 36 cans has a mean amount of 12.19 ounces. The distribution of sample means of size n = 36 is normal with an assumed mean of 1
> The U.S. Food and Drug Administration plans to conduct a survey to identify the average (mean) amount of arsenic found in the rice grown on farms located in Arkansas. Arkansas has about 3000 rice farms. a. What would be a practical problem of using a str
> On days of presidential elections, the news media organize an exit poll in which specific polling stations are randomly selected and all voters are surveyed as they leave the premises.
> When 50 adult females were randomly selected and their pulse rates were measured, the mean of 75.5 beats per minute (bpm) was obtained. When the sample size was increased to 147 adult females, the mean of 74.0 bpm was obtained. Is the mean from the large
> In a clinical trial, 11,000 male physicians were treated with aspirin and another 11,000 male physicians were given a placebo. A variable of interest was the proportion of heart attacks among the physicians. Do the results from this study apply to female
> Among all of the 50,000 aspirin tablets produced by a pharmaceutical company over a given period, a sample of 200 tablets is tested and the mean amount of aspirin in these tablets is found to be 328 milligrams (mg) with a standard deviation of 12 mg.
> Among all of the 50,000 aspirin tablets produced by a pharmaceutical company over a given period, a sample of 200 tablets is tested and it is found that 4% of them do not meet specifications.
> A college has 3427 enrolled students. When 50 of them were randomly selected for a survey, it was found that 10% of them were in favor of fees for parking permits.
> For a random sample of 575 randomly selected car batteries, it was found that their output had a mean of 12.2 volts and a standard deviation of 1.4 volts.
> Our study measured the birth weights and incidence of jaundice among a sample of babies born at our hospital, and we found x‾ = 6.7 pounds and p̂ = 0.45, or 45% showed signs of jaundice.
> Although Nielsen surveys only a few thousand households out of the millions that own TVs, they have a good chance of getting an accurate estimate of the proportion of the population watching the Super Bowl.
> Nielsen Media Research determined the precise proportion of all Americans watching the Super Bowl by conducting a survey of a few thousand households.
> a. Is the elevation change more from B to D or from D to F? b. What is the elevation change if you walk from A to C to D to A? Figure 3.27
> Refer to the Dvorak data in Exercise 21 in Section 3.1 and construct a dot plot. Compare the result to the dot plot from Exercise 21 above. Based on the results, does either keyboard configuration appear to be better? Explain.
> This section includes formulas using the symbols μ, and n. What do these symbols represent?
> Police set up a sobriety checkpoint at which every fifth driver is stopped and interviewed.
> One study of heart disease involved treating male physicians with daily doses of aspirin. Because the study concluded that aspirin helps males avoid heart disease, it follows that females can also avoid heart disease by taking aspirin.
> In designing new desks for an incoming class of 25 kindergarten girls, an important characteristic of the desks is that they must accommodate the sitting heights of those students. (The sitting height is the height measured from the bottom of the feet, w
> Federal Aviation Administration rules require airlines to estimate the weight of a passenger as 195 pounds, including carry-on baggage. Men have weights (without baggage) that are normally distributed with a mean of 172 pounds and a standard deviation of
> Currently, quarters have weights that are normally distributed with a mean of 5.670 grams and a standard deviation of 0.062 gram. A vending machine is configured to accept only those quarters with weights between 5.550 grams and 5.790 grams. a. If 280 di
> M&M plain candies have weights that are normally distributed with a mean weight of 0.8565 gram and a standard deviation of 0.0518 gram (based on measurements taken by one of the authors of this text). A random sample of 100 M&M candies is obtained from a
> When women first became pilots of fighter jets, engineers needed to redesign the ejection seats because they had been designed for men only. The ACES-II ejection seats were designed for men weighing between 140 pounds and 211 pounds. The population of wo
> Engineers must consider the breadths of male heads when designing motorcycle helmets for men. Men have head breadths that are normally distributed with a mean of 6.0 inches and a standard deviation of 1.0 inch (based on anthropometric survey data from Go
> a. If you walk from A to C, do you walk uphill or downhill? b. If you walk directly from E to F, does your elevation increase, decrease, or remain the same? Figure 3.27
> An aircraft strobe light is designed so that the times between flashes are normally distributed with a mean of 3.00 seconds and a standard deviation of 0.40 second. a. What is the likelihood (percentage) that an individual time between flashes is greater
> Assume that cans of cola are filled so that the actual amounts are normally distributed with a mean of 12.00 ounces and a standard deviation of 0.11 ounce. a. What is the likelihood (percentage) that a sample of 49 cans will contain a mean amount of at l
> What percentage of individual adult females have weights between 74 kg and 80 kg? If samples of 36 adult females are randomly selected and the mean weight is computed for each sample, what percentage of sample means is between 74 kg and 80 kg?
> In phase II testing of a new drug designed to increase the red blood cell count, a researcher obtains envelopes with the names and addresses of all treated subjects. She wants to increase the dosage in a sub-sample of 12 subjects, so she thoroughly mixes
> What percentage of individual adult females have weights between 75 kg and 81 kg? If samples of 100 adult females are randomly selected and the mean weight is computed for each sample, what percentage of sample means is between 75 kg and 81 kg?
> What percentage of individual adult females have weights greater than 79 kg? If samples of 25 adult females are randomly selected and the mean weight is computed for each sample, what percentage of the sample means are greater than 79 kg?
> What percentage of individual adult females have weights less than 75 kg? If samples of 36 adult females are randomly selected and the mean weight is computed for each sample, what percentage of the sample means are less than 75 kg?
> Rolling a fair 10-sided die produces a uniformly distributed set of numbers between 1 and 10 with a mean of 5.5 and a standard deviation of 2.872. Assume that n such dice are rolled many times and the mean of the n outcomes is computed each time. a. Find
> Rolling a fair 12-sided die produces a uniformly distributed set of numbers between 1 and 12 with a mean of 6.5 and a standard deviation of 3.452. Assume that n such dice are rolled many times and the mean of the n outcomes is computed each time. a. Find
> Weights of adult males are normally distributed with a mean of 85.5 kg and a standard deviation of 17.7 kg. Assume that many samples of size n are taken from a large population of adult males and the mean weight is computed for each sample. a. If the sam
> Are there any locations that stand out as unusual and that might therefore warrant special study? Explain Figure3.26
> IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Assume that many samples of size n are taken from a large population of people and the mean IQ score is computed for each sample. a. If the sample size is n = 64, find
> If we record the means from 1000 rolls of 100 dice, the resulting distribution will be closer to a normal distribution than if we record the means from 1000 rolls of 10 dice.
> For the samples described in Exercise 6, the sample means will vary less than the original incomes.
> The distribution of incomes of adults is a right- skewed distribution. Therefore, if we select many samples of 64 incomes at random from this distribution, the means of these samples will also have a right-skewed distribution.
> The Journal of the American Medical Association prints an article evaluating a drug, and some of the physicians who wrote the article received funding from the pharmaceutical company that produces the drug.
> A process consists of repeating this operation: Randomly select two values from a normally distributed population and then find the mean of the two values. The sample means will be normally distributed, even though each sample has only two values.
> Briefly explain a major reason why the Central Limit Theorem is useful in statistics.
> What is the Central Limit Theorem? When does it apply?
> My professor graded the final on a curve, and she gave a grade of A+ to anyone who had a standard score of 2 or more.
> What is a percentile? Describe how Table A-1 (in Appendix A) allows you to relate standard scores and percentiles.
> Are there broad regions where melanoma mortality is more common than others? Which ones, and what do they have in common? Figure 3.26
> What is a standard score? How do you find the standard score for a particular data value?
> For a normal distribution, approximately what fraction of data values lie more than 1 standard deviation above the mean? When might data values be considered unusual? Explain.
> What is the 68-95-99.7 rule for normal distributions? Does it apply to other (not normal) distributions as well?
> Based on a random sample of movie lengths, the mean length is 110.5 minutes with a standard deviation of 22.4 minutes. Assume that movie lengths are normally distributed. a. What percentage of movies are more than 2 hours long? b. What percentage of movi
> Heights of adult American males are normally distributed with a mean of 69.0 inches and a standard deviation of 2.8 inches. The U.S. Marine Corps requires that males have heights between 64 inches and 78 inches. What percentages of males are eligible for
> Monsanto hires independent university scientists to determine whether its new, GMO (genetically modified organism) soybean poses any threat to consumers.
> At the district spelling bee, the girls have normally distributed scores with a mean score of 71 points and a standard deviation of 6 those students with a score greater than 75 are eligible to go to the state spelling bee. What percentage of the girls w
> Assume that body temperatures of healthy adults are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F. a. What percentage of adults have body temperatures greater than 99.99°F? b. What percentage of adults have body temperatu
> Assume that the scores on the Graduate Record Exam (GRE) are normally distributed with a mean of 497 and a standard deviation of 115. a. A graduate school requires a GRE score of 650 for admission. To what percentile does this correspond? b. A graduate
> Based on data from the College Board, SAT scores on the Math Level 1 test are normally distributed with a mean of 621 and a standard deviation of 96. a. Find the percentage of scores greater than 600. b. Find the percentage of scores less than 700. c.
> The stack plot in Figure 3.25 shows the numbers of bachelor’s degrees awarded to males and females since 1970. The last few years are projections. a. Estimate the numbers of bachelor’s degrees to be awarded to males an
> Lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. a. What percentage of pregnancies last less than 250 days? b. What percentage of pregnancies last more than 300 days? c. If a birth is considered
> Consider the following table, showing the official mean weight and estimated standard deviation for five U.S coins. Suppose a vending machine is designed to reject all coins with weights more than 2 standard deviations above or below the mean. For each c
> The percentage of heights between 150 cm and 170 cm
> The percentage of heights between 180 cm and 200 cm
> The percentage of heights between 160 cm and 188 cm
> The percentage of heights between 167 cm and 181 cm
> Consumer Reports magazine prints a review of new cars and does not accept free products or run any advertisements from any companies.
> The percentage of heights greater than 180 cm
> The percentage of heights less than 146 cm
> The percentage of heights less than 200 cm
> The stack plot in Figure 3.24 on the next page shows Congressional Budget Office data for actual spending (through 2011) and projected spending on federal entitlement programs through 2085 as percentages of the gross domestic product (GDP). Interpret the
