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Mind On Statistics 4th Edition David D Busch, Jessica M Utts, Robert F Heckard - Solutions
Refer to Exercise 9.78, in which 25 participants took zinc lozenges and 23 participants took placebo lozenges.Suppose that, in truth, if everyone in the population took zinc lozenges, symptoms would last an average of 4.5 days with a standard deviation of 1.5 days, while if they took placebo
Volunteers who had developed a cold within the previous 24 hours were randomized to take either zinc or placebo lozenges every 2 to 3 hours until their cold symptoms were gone (Prasad et al. 2000). Twenty-five participants took zinc lozenges, and 23 participants took placebo lozenges. The mean
In situations for which the parameter of interest is the difference in population means for independent samples, is it necessary for the two samples sizes n1 and n2 to be equal in order for the sampling distribution of the sample statistic to be approximately normal? If not, what condition is
Is the mean of the sampling distribution of x1 2 x2 always 0?If so, explain why. If not, explain what condition(s) would be necessary for it to be 0.
Horne, Reyner, and Barrett (2003) describe a study in which 12 men took a simulated driving test after drinking alcohol at lunch, getting too little sleep the night before, neither(control condition), or both. Each of the men took the test under all four conditions. One of the measurements taken
In a project for a statistics class, a group of students decided to compare a name brand (Brand 1) and a store brand(Brand 2) of microwave popcorn to see which one left fewer unpopped kernels, on average, after the recommended popping time. They knew that microwave ovens can differ, so they asked
Is the mean of the sampling distribution of d always 0? If so, explain why. If not, explain what it means about the population when it is 0.
Explain which parameter is being described in each of the following situations: m, md, or m1 2 m2.a. The mean grade point average for all students at your school.b. The difference in the mean age at marriage of women who were married for the first time in 1950 and women who were married for the
Explain which parameter is being described in each of the following situations: m, md, or m1 2 m2.a. The difference in the mean number of pushups new male and female military recruits can do.b. The mean change in the number of pushups female recruits can do at the end of basic training, compared
Thirteen individuals each take a strength test both before and again after they participate in a 6-week weight-training program. The investigator wants to estimate the average gain in strength due to the training program. Does this comparison involve paired data or data from two independent samples?
Twenty sophomores are randomly selected from campus dormitories at a college, and 20 other sophomores are randomly selected from students who live off campus. The mean grade point averages of the two groups are compared.Does this comparison involve paired data or data from two independent samples?
A rental car company has noticed that the distribution of the number of miles customers put on rental cars per day is somewhat skewed to the right, with an occasional high outlier. The distribution has a mean of 80 miles and a standard deviation of 50 miles.a. Draw a picture of a normal curve with
Explain the difference between the standard deviation of the sampling distribution of the mean and the standard error of the mean. Explain which one is more likely to be used in practice and why.
Describe the sampling distribution for the statistic of interest in each of the following stories, including numerical values for the mean and standard deviation of the sampling distribution.a. A fisherman takes tourists out fishing and has noticed that over the long run, the weights of a certain
Small planes cannot fly well if the payload (people, luggage, and fuel) weighs too much. Suppose that an airline runs a commuter flight that holds 40 people. The airline knows that the weights of passenger plus luggage for typical customers on this flight is approximately normal with a mean of 210
Vehicle speeds at a certain highway location are believed to have approximately a normal distribution with mean m 5 60 mph and standard deviation s 5 6 mph. The speeds for a randomly selected sample of n 5 36 vehicles will be recorded.a. Give numerical values for the mean and standard deviation of
◆ The cholest dataset on the companion website includes cholesterol levels measured for 28 heart attack patients 2 days after their attacks. The sample mean for this sample is 253.9, and the standard deviation is 47.7.a. Assuming that this was a sample from a larger population of heart attack
A randomly selected sample of n 5 60 individuals over 65 years old takes a test of memorization skills. The sample mean is x 5 53, and the standard deviation is s 5 7.2. Give the numerical value of the standard error of the mean.
The weights of men in a particular age group have mean m 5 170 pounds and standard deviation s 5 24 pounds.a. For randomly selected samples of n 5 16 men, what is the standard deviation of the sampling distribution of possible sample means?b. For randomly selected samples of n 5 64 men, what is the
hours and the standard deviation was s 5 1.75 hours. Consider randomly selected samples of n 5 190 students.a. What is the value of the mean of the sampling distribution of possible sample means?
Example 9.2 (p. 321) concerned selecting random samples of n 5 190 from a population of student responses to the question “How many hours of sleep did you get last night?”Suppose that in a large population of students, the mean amount of sleep the previous night was m 5
Explain whether one (or both) of the situations for the approximate normality of the sampling distribution of the mean holds in each scenario.a. Mean normal body temperature will be determined for a randomly selected sample of 18 individuals. In the population of all humans, normal body temperature
Refer to Exercises 9.56 and 9.57. Which of the following could be different if two different sets of samples of 900 children are taken? Explain.a. The value of the mean of the sampling distribution of p^ 1 2 p^ 2.b. The approximate shape of the sampling distribution of p^ 1 2 p^ 2.c. The standard
Refer to Exercise 9.56, including the information given in part (c). Which of the following could be specified before the samples are taken? Explain.a. The value of the mean of the sampling distribution of p^ 1 2 p^ 2.b. The approximate shape of the sampling distribution of p^ 1 2 p^ 2.c. The
Researchers would like to know whether the proportions of elementary school children who are obese differ in rural and urban areas. They plan to take random samples of 900 children from urban areas and 900 children from rural areas and compare the proportions that are obese. Define Population 1 to
When taking large random samples from two different populations and finding the sampling distribution of p^ 1 2 p^ 2 explain which of the following would change if the sample sizes are changed.a. The mean of the sampling distribution.b. The approximate shape of the sampling distribution.c. The
There is some evidence that women tend to vote earlier in the day and men tend to vote later in the day. If men and women vote differently, then exit polls taken early in the day may not reflect the eventual results. Suppose that in a particular election, 51% of the voters who vote in the morning
For each of the following situations, explain whether the conditions hold for the sampling distribution of p^ 1 2 p^ 2 to be approximately normal:a. A polling organization wants to compare the proportions of men and women who favor a particular candidate in an upcoming election and plans to take
Refer to Exercises 9.50 and 9.51. The situations are repeated here. In each case, draw a picture of the sampling distribution of p^ 1 2 p^ 2, similar to Figure 9.5 (p. 331).a. The populations are male and female registered voters in the United States, and it is of interest to compare the
Refer to Exercise 9.50. In each part, specify the standard deviation of the sampling distribution of the difference in sample proportions.
For each of the following situations, specify the mean of the sampling distribution of the difference in sample proportions:a. The populations are male and female registered voters in the United States, and it is of interest to compare the proportions of males and females who are registered with
If the population proportions are equal, what will be the mean of the sampling distribution of p^ 1 2 p^ 2?
A student commented, “Once a poll has been taken, p^ (the sample proportion with a certain opinion) is a known value, so it cannot have a distribution of possible values. Therefore, I don’t understand what you mean by the sampling distribution of p^.” Write an explanation for this student.
Explain the difference between the standard error of p^ and the standard deviation of the sampling distribution of p^.Write down the formula for each one. Which one is more likely to be used in practice? Why?
Refer to Exercise 9.45, in which 20% of the patients in a medical clinic are obese. If the clinic took repeated random samples of 100 observations and found the sample proportion who were obese, into what interval should those sample proportions fall about 95% of the time?
Recent studies have shown that about 20% of American adults fit the medical definition of being obese. A large medical clinic would like to estimate what percentage of their patients are obese, so they take a random sample of 100 patients and find that 18 are obese. Suppose that in truth, the same
For each of the following, if the scenario fits the conditions for the approximate normality of the sampling distribution for a sample proportion, describe the sampling distribution of p^ and draw a picture of it. If the scenario doesn’t fit, explain why not (in which case you do not have the
Refer to Exercise 9.42, in which p^ is the sample proportion of 400 randomly selected club members who require aid in a 12-month period. Based on the Empirical Rule, into what interval should p^ fall about 95% of the time?
An automobile club comes to the aid of stranded motorists who are members. Over the long run, about 5% of the members utilize this service in any 12-month period. A small town has 400 members of the club. Consider the 400 members to be representative of the larger club membership. Let p^ be the
Suppose medical researchers think that p 5 .70 is the proportion of all teenagers with high blood pressure whose blood pressure would decrease if they took calcium supplements. To test this theory, the researchers plan a clinical trial (experiment) in which n 5 200 teenagers with high blood
Suppose the probability is p 5 .2 that a person who purchases an instant lottery ticket wins money, and this probability holds for every ticket purchased. Consider different random samples of n 5 64 purchased tickets. Let p^ 5 sample proportion of winning tickets in a sample of 64 tickets.a. Give
In a random sample of n 5 500 adults, 300 individuals say that they believe in love at first sight.a. Calculate the value of p^ 5 sample proportion that believes in love at first sight.b. Calculate the standard error of p
In a random sample of 1000 adults aged 18 to 25, 590 individuals said that they drink alcohol at least once a month.a. Calculate the value of p^ 5 sample proportion that drinks alcohol at least once a month.b. Calculate the standard error of p^.
A polling organization polls n 5 100 randomly selected registered voters to estimate the proportion of a large population that intends to vote for Candidate Y in an upcoming election. Although it is not known by the polling organization, p 5 .55 is the actual proportion of the population that
In a random sample of n 5 200 drivers, 50 individuals say that they never wear a seatbelt when driving.a. Give a numerical value of p^ 5 sample proportion that never wears a seatbelt when driving.b. Calculate the standard error of p^c. In a different survey of n 5 400 drivers, 100 people say they
Refer to Exercises 9.33 and 9.34, and note that for both exercises the value of n in part (b) is four times what it is in part(a), while p is the same. Use the answers to Exercise 9.33 or 9.34 to describe how changing the sample size affects the mean and standard deviation of the distribution of
Calculate the mean and the standard deviation of the sampling distribution of possible sample proportions for each combination of sample size (n) and population proportion (p).a. n 5 64, p 5 .8.b. n 5 256, p 5 .8.
Calculate the mean and the standard deviation of the sampling distribution of possible sample proportions for each combination of sample size (n) and population proportion(p).a. n 5 400, p 5 .5.b. n 5 1600, p 5 .5.
(p. 321), which described the sampling distribution of the mean of 190 sleep values, explain why this result makes sense.b. Consider the extreme case in which everyone in the population is measured. In that case, what would be the standard deviation for the sampling distribution of the sample mean?
One result stated in Section 9.3 is that the standard deviation of the sampling distribution gets smaller as the sample size gets larger.a. Using the context of Example
Refer to the smooth curve in Figure 9.1 (p. 321), illustrating the sampling distribution of mean hours of sleep for a sample of 190 college students, taken from a population of sleep hours with a mean of 7.1 hours and standard deviation of 2 hours. Explain whether each of the following would be
Refer to Exercises 9.28 and 9.29. Using that context as an example, define in your own words what is meant by a sampling distribution of the sample mean.
Refer to Exercise 9.28. If a different sample of 100 marathon runners was taken from the same population, which of the following would be the same as it was for the first sample, and which would be different? Explain your answer in each part.a. The value of the mean of the sampling distribution of
Researchers would like to know the mean body temperature of runners at the completion of a marathon. They plan to take a random sample of 100 runners who complete a marathon and record their temperatures. Which of the following will the researchers know after they have taken the sample?Explain your
Consider a situation in which researchers measure a random sample of 50 newborn babies and the average weight is calculated. If a new random sample of 50 newborns is taken from the same population, explain whether each of the following would change:a. The population mean, m.b. The sample mean, xc.
Consider a situation in which a random sample of 1000 adults is surveyed and the proportion that primarily buys organic vegetables is found. If a new random sample of 1000 adults is taken from the same population, explain whether each of the following would change:a. The population proportion, p.b.
Explain what symbol (p^, m, etc.) would be used to represent each of the following:a. The mean of the sampling distribution of p^.b. One value from the sampling distribution of p^.
Explain what symbol (p^, m, etc.) would be used to represent each of the following:a. The mean of the sampling distribution of the sample mean.b. One value from the sampling distribution of the sample mean.
Consider a situation in which a sample of 100 SAT scores is taken from the population of all people who took the test in a given year. Which would be more spread out: the sampling distribution of the sample mean for 100 SAT scores or a histogram of the 100 SAT scores reported in one sample?Explain.
In Example 9.1 (p. 316), we discussed a few different parameters that might be of interest in exploring the question of whether first-year college students gain weight. In each of the following cases, explain what population(s) the sample(s) should be drawn from and what variable or variables would
For each of the following situations from earlier chapters, define the parameter of interest and the sample statistic that would be used to estimate that parameter, choosing from among the five described in Section 9.2. When defining the parameter, make sure you explain what constitutes the
Give an example of a research question for which the population parameter of interest would be m1 2 m2.
Give an example of a research question for which the population parameter of interest would be p1 2 p2.
For each of the following research questions, would it make more sense to collect independent samples or paired data?Explain.a. In the United States, on average, what is the age difference between husbands and wives?b. What is the difference in average ages at which teachers and plumbers retire?c.
Refer to Exercise 9.16. Does the situation described represent independent samples or paired data? Explain.
A sample of 100 students at a university was asked how many hours a week they spent studying and how many they spent socializing. The difference was computed for each student. What are the research question, the population parameter, and the sample estimate for this study?
A study was done by randomly assigning 200 volunteers with sore throats to either drink a cup of herbal tea or use a throat lozenge to ease their pain. The percentages reporting relief for the two methods were compared. What are the research question, the population parameter, and the sample
A medical researcher wants to estimate the difference in the proportions of women with high blood pressure for women who use oral contraceptives versus women who do not use oral contraceptives. In an observational study involving a sample of 900 women, the researcher finds that .15 (15%) of
A student doing a project for a statistics class wants to estimate the difference between the mean number of music CDs owned by male and female students. He surveys 50 male students and 45 female students about the number of music CDs they own. The mean number for the males is 110, and the mean
A large Internet provider conducted a survey of its customers. One question that it asked was how many e-mail messages the respondent had received the previous day. The mean number was 13.2.a. What is the research question of interest for this survey?b. What is the population parameter in this
In a random sample of 300 parents in a school district, 104 supported a controversial new education program. The purpose of the survey was to estimate the proportion of all parents in the school district who support the new program.a. What is the research question of interest for this survey?b.
Explain whether x or m is the correct statistical notation for each mean described.a. The mean hours of study per week was 15 hours for a sample of n 5 30 students at a college.b. A university administrator determines that the mean age of all students at a college is 20.2 years.
Explain whether p^ or p is the correct statistical notation for each proportion described:a. The proportion that smokes in a randomly selected sample of n 5 300 students in the eleventh and twelfth grades.b. The proportion that smokes among all students in the eleventh and twelfth grades in the
In Section 9.1, read the paragraph that begins with“Hypothesis testing is applied to population parameters by specifying a null value...” on page 314. Now refer to Exercise 9.6, for which the population parameter of interest would be the difference in mean salaries for all males and all females
Refer to Exercise 9.6. Suppose that the 100 men and 100 women in the sample were all of the company’s employees.Now answer questions a andb, repeated here:a. Are the mean salaries in this example statistics or parameters? Explain.b. Based on these means, can the shareholders determine that the
Some stockholders want to know whether the mean salary for male employees in a large company is higher than the mean salary for female employees. The company allows them access to salary information for a random sample of 100 male and 100 female employees, and the mean salaries are $51,000 for the
A report in Newsweek about the relationship among diet, genes, and disease (Underwood and Adler, 2005) discussed the E4 allele for a protein known as Apo E and said “an estimated 15 to 30 percent of the population has at least one copy of the allele.” People who have the allele are at higher
Part (a) of this exercise is also Exercise 1.6. Using Case Study 1.6 on pages 5–6 as an example:a. Explain the difference between a population and a sample.b. Use the proportion of placebo takers who had heart attacks, which was .017, to explain the difference between a population parameter and a
As we learned in Chapter 5, a 95% confidence interval is an interval such that we are 95% confident it covers the truth.Does “the truth” in this definition refer to the truth about a sample statistic or about a population parameter? Or does it depend on the situation? Explain.
A polling organization plans to sample 1000 adult Americans to estimate the proportion of Americans who think crime is a serious problem in this country. In the context of this poll, explain the difference between a statistic and a parameter.What is the parameter of interest to the polling
In each situation, explain whether the value given in bold print is a statistic or a parameter:a. A polling organization samples 1000 adults nationwide and finds that 72% of those sampled favor tougher penalties for persons convicted of drunk driving.b. In the year 2000 census, the U.S. Census
New spark plugs have just been installed in a small airplane with a four-cylinder engine. There is one spark plug per cylinder, so four spark plugs have been installed. For each spark plug, the probability that it is defective and will fail during its first 20 minutes of flight is 1/10,000,
The vehicle speeds at a particular interstate location can be described by a normal curve. The mean speed is 67 mph, and the standard deviation is 6 mph. What is the proportion of vehicle speeds at this location that are faster than 75 mph?
Answer Thought Question 8.4 on page 273.
You will need a computer program that simulates a large number of binomial random variables at once for this exercise. Refer to Example 8.34 (p. 300), “Strategies for Studying When You Are out of Time.” Simulate what would happen if you used Strategy 2, in which the total score was the sum of
Use the rule referenced from Chapter 7 to verify that the relationship is correct for a standard normal random variable Z:a. Use Rule 1 of Chapter 7 to verify that P(Z .a) 5 1 2 P(Z # a).b. Use Rule 2b of Chapter 7 to verify that P(Z # b)5 P(Z #a) 1 P(a # Z # b). Use that result to show that P(a #
Suppose that mean body temperature for adults has a normal distribution with mean μ 5 98.2 degrees Fahrenheit and standard deviation s 5 0.8 degrees.a. What is the probability that a randomly selected adult has a body temperature less than or equal to 98.6 degrees?b. What is the probability that a
Shaun (3 years old) and Patrick (4 years old) are each allowed to pick one book for bedtime stories, and the parent on duty reads the two books sequentially. The times for reading (single) books in the collection from which they select are approximately normally distributed, with a mean of 5
Kim and her sister Karen each plan to have four children.Assume that the probability of a girl is .50, independent across births.a. If X 5 number of girls Kim will have, what is the distribution of X?b. If Y 5 number of girls Karen will have, what is the distribution of Y?c. What is the
Assume that right handspan measurements for college women are approximately normally distributed, with a mean of 20 cm and a standard deviation of 1.8 cm, and that for men, they are normally distributed, with a mean 22.5 cm and a standard deviation of 1.5 cm. Measure your own right handspan.a.
The standard medical treatment for a certain disease is successful in 60% of all cases.a. The treatment is given to n 5 200 patients. What is the probability that the treatment is successful for 70% or more of these 200 patients? (Hint: 70% of n 5 200 is 140 patients.)b. The treatment is given to
Use the direct formulas for expected value and standard deviation to verify that the mean and standard deviation for a binomial random variable with n 5 2 and p 5 .5 are μ 5 1 and s 5 .7071, respectively. (Hint: The only possible values of X are 0, 1, 2. Find their probabilities and use the
Explain which of the conditions for a binomial experiment is not met for each of the following random variables:a. A ten-question quiz has five true–false questions and five multiple-choice questions. Each multiple-choice question has four possible choices for the answer. A student randomly picks
Burt pays $30 a year for towing insurance. He thinks that the probability that he will need to have his car towed once in the next year is 1/10 and the probability that he will have to have it towed more than once is zero. It will cost $100 if his car is towed if he doesn’t have insurance but
Refer to Exercise 8.100. What is the mean number of courses taken by full-time students?
Suppose that a college determines the following distribution for X 5 number of courses taken by full-time students this semester:
In the casino game of roulette, a gambler can bet on which of 38 numbers will be the result when the roulette wheel is spun. On a $2 bet, a gambler gains $70 if he or she picks the right number but loses the $2 otherwise.a. Let X 5 amount gained or lost on a $2 bet on a roulette number. Write out
Suppose that 10% of a population is left-handed. What is the probability that in a sample of n 5 10 individuals, 3 or more individuals are left-handed?
A histogram is drawn of the weights of all students in a class of 100 men and 100 women. Would this histogram have a bell shape? Briefly explain your answer.
Do you think that a normal curve would be a good approximation to the distribution of the ages of all individuals in the world? Briefly explain your answer.
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