Answer: z Scores. In Exercises, express allz scores with

Chapter 3.4,Problem 1BSC z Scores James Madison, the fourth President of the United States, was 163 cm tall. His height converts to the z score of –2.28 when included among the heights of all presidents. Is his height above or below the mean? How many standard deviations is Madison’s height away from the mean?

Find the mean of these times that American Airlines flights used to taxi to the Los Angeles terminal after landing from a flight: 12 min, 8 min, 21 min, 17 min, 12 min. (Data are from Data Set 15 in Appendix B.)

Designing Gloves An engineer is designing a machine to manufacture gloves and she obtains the following sample of hand lengths (mm) of randomly selected adult males (based on anthropometric survey data from Gordon, Churchill, et al.): a. Are exact hand lengths from a population that is discrete or continuous? b. What is the level of measurement of the hand lengths? (nominal, ordinal, interval, ratio)

Ergonomics When designing an eye-recognition security device, engineers must consider the eye heights of standing women. (It's easy for men to bend lower, but it's more difficult for women to rise higher.) Listed below are the eye heights (in millimeters) obtained from a simple random sample of standing adult women (based on anthropometric survey data from Gordon, Churchill, et al.). Use the given eye heights to find the (a) mean? (b) median? (c) mode? (d) midrange? (e) range? (f) standard deviation? (g) variance? (h) Q 1 ? (i) Q 3 .

Problem 2BSC z Scores If your score on your next statistics test is converted to a z score, which of these zscores would you prefer: ?2.00, ?1.00, 0, 1.00, 2.00? Why?

What is the median of the sample values listed in Exercise 1?

Frequency Distribution Use the hand lengths in Exercise 1 to construct a frequency distribution. Use a class width of \(10 \mathrm{~mm}\), and use \(150 \mathrm{~mm}\) as the lower class limit of the first class. Equation Transcription: Text Transcription: 10 mm 150 mm

z Score Using the sample data from Exercise 1, find the z score corresponding to the eye height of \(1642 \mathrm{~mm}). Is that eye height unusual? Why or why not? Equation Transcription: Text Transcription: 1642 mm

Boxplots Shown below is a STATDISKgenerated boxplot of the amounts of money (in millions of dollars) that movies grossed (based on data from the Motion Picture Association of America). What do the displayed values of 5, 47, 104, 121, and 380 tell us?

What is the mode of the sample values listed in Exercise 1?

Histogram Use the frequency distribution from Exercise 2 to construct a histogram.

Boxplot Using the same standing heights listed in Exercise 1, construct a boxplot and include the values of the 5-number summary. Does the boxplot indicate that the data are from a population with a normal (bell-shaped) distribution? Explain.

Problem 4BSC Measures of Location The values of P50, Q2, and the median are found for the net incomes reported on all individual 1040 tax forms filed last year. What do those values have in common?

The standard deviation of the sample values in Exercise 1 is 5.0 min. What is the variance (including units)?

Stemplot Use the hand lengths from Exercise 1 to construct a stemplot.

Problem 4RE ZIP Codes An article in the New York Times noted that these new ZIP codes were created in New York City: 10065, 10021, 10075. Find the mean of these three numbers. What is fundamentally wrong with this result?

Problem 5BSC z Scores. In Exercises, express allz scores with two decimal places. Obama’s Net Worth As of this writing, Barack Obama is President of the United States and he has a net worth of $3,670,505. The 17 members of the Executive Branch have a mean net worth of $4,939,455 with a standard deviation of $7,775,948 (based on data from opensecrets.org). a. What is the difference between President Obama’s net worth and the mean net worth of all members of the Executive Branch? ________________ b. How many standard deviations is that (the difference found in part (a))? ________________ c. Convert President Obama’s net worth to a z score. ________________ d. If we consider “usual” amounts of net worth to be those that convert to z scores between ?2 and 2, is President Obama’s net worth usual or unusual?

Problem 5CQQ The taxi-in times for 48 flights that landed in Los Angeles have a mean of 11.4 min and a standard deviation of 7.0 min. What is the z score for a taxi-in time of 6 min?

Earthquakes Data Set 16 in Appendix B lists 50 magnitudes (Richter scale) of 50 earthquakes, and those earthquakes have magnitudes with a mean of 1.184 with a standard deviation of 0.587. The strongest of those earthquakes had a magnitude of 2.95. a. What is the difference between the magnitude of the strongest earthquake and the mean magnitude? b. How many standard deviations is that (the difference found in part (a))? c. Convert the magnitude of the strongest earthquake to a z score. d. If we consider “usual” magnitudes to be those that convert to z scores between ? 2 and 2, is the magnitude of the strongest earthquake usual or unusual?

Descriptive Statistics Use the hand lengths in Exercise 1 and find the following: (a) mean? (b) median? (c) standard deviation? (d) variance? (e) range. Include the appropriate units of measurement.

Comparing BMI The body mass indices (BMI) of a sample of males have a mean of 26.601 and a standard deviation of 5.359. The body mass indices of a sample of females have a mean of 28.441 and a standard deviation of 7.394 (based on Data Set 1 in Appendix B). When considered among members of the same gender, who has the relatively larger BMI: a male with a BMI of 28.00 or a female with a BMI of 29.00? Why?

Problem 6CQQ You plan to investigate the variation of taxi-in times for flights that have landed in Los Angeles. Name at least two measures of variation for those data.

Normal Distribution Instead of using the hand lengths in Exercise 1, a much larger sample of hand lengths is used and a frequency distribution is created. The frequencies listed in order are 1, 8, 56, 237, 382, 228, 48, 4, 1. Does it appear that the sample is from a population having a normal distribution? Explain.

Problem 6RE Movies: Estimating Mean and Standard Deviation Consider the prices of regular movie tickets (not 3-D, and not discounted for children or seniors). The prices of movie tickets are $15, $20, $22. $24 and $25. a. Estimate the mean price. ________________ b. Use the range rule of thumb to make a rough estimate of the standard deviation of the prices.

Problem 7CRE Sampling Shortly after the World Trade Center towers were destroyed, America Online ran a poll of its Internet subscribers and asked this question: “Should the World Trade Center towers be rebuilt?” Among the 1,304,240 responses, 768,731 answered “yes,” 286,756 answered “no,” and 248,753 said that it was “too soon to decide.” Given that this sample is extremely large, can the responses be considered to be representative of the population of the United States? Explain.

Problem 7BSC z Scores. In Exercises, express allz scores with two decimal places. Jobs’ Job When Steve Jobs was Chief Executive Officer (CEO) of Apple, he earned an annual salary of $1. The CEOs of the 50 largest U.S. companies had a mean salary of $1, 449, 779 and a standard deviation of $527, 651 (based on data from USA Today). a. What is the difference between Jobs’ salary and the mean CEO salary? ________________ b. How many standard deviations is that (the difference found in part (a))? ________________ c. Convert Steve Jobs’ salary to a z score. ________________ d. If we consider “usual” salaries to be those that convert to z scores between ?2 and 2, is Steve Jobs’ salary usual or unusual?

Problem 7CQQ Consider a sample taken from the population of all taxi-in times for all flights that land in Los Angeles. Identify the symbols used for the sample mean and the population mean.

Problem 7RE Professors: Estimating Mean and Standard Deviation Use the range rule of thumb to estimate the standard deviation of ages of all teachers at your college.

Student's Pulse Rate A male student of the author has a measured pulse rate of 52 beats per minute. Based on Data Set 1 in Appendix B, males have a mean pulse rate of 67.3 beats per minute and a standard deviation of 10.3 beats per minute. a. What is the difference between the student's pulse rate and the mean pulse rate of males? b. How many standard deviations is that (the difference found in part (a))? c. Convert the student's pulse rate to a z score. d. If we consider “usual” pulse rates to be those that convert to z scores between ? 2 and 2, is the student's pulse rate usual or unusual?

Problem 8CQQ Consider a sample taken from the population of all taxi-in times for all flights that land in Los Angeles. Identify the symbols used for the sample standard deviation, the population standard deviation, the sample variance, and the population variance.

Histogram The accompanying histogram depicts outcomes of digits from the California Daily 4 lottery. What is the major flaw in this histogram?

Problem 8RE Aircraft Design Engineers designing overhead bin storage in an aircraft must consider the sitting heights of male passengers. Sitting heights of adult males have a mean of 914 mm and a standard deviation of 36 mm (based on anthropometric survey data from Gordon, Churchill, et al.). Use the range rule of thumb to identify the minimum “usual” sitting height and the maximum “usual” sitting height. Which of those two values is more relevant in this situation? Why?

Problem 9BSC Usual and Unusual Values. In Exercises, consider a value to be unusual if its z score is less than ?2 or greater than 2. IQ Scores The Wechsler Adult Intelligence Scale measures IQ scores with a test designed so that the mean is 100 and the standard deviation is 15. Consider the group of IQ scores that are unusual. What are the z scores that separate the unusual IQ scores from those that are usual? What are the IQ scores that separate the unusual IQ scores from those that are usual?

Problem 9CQQ Approximately what percentage of taxi-in times is less than the 75th percentile?

Interpreting a Boxplot Shown below is a boxplot of a sample of 20 brain volumes \(((\operatorname{cm} 3)\). What do the numbers in the boxplot represent? Equation Transcription: Text Transcription: ( cm 3)

Problem 10BSC Usual and Unusual Values. In Exercises, consider a value to be unusual if its z score is less than ?2 or greater than 2. Designing Aircraft Seats In the process of designing aircraft seats, it was found that men have hip breadths with a mean of 36.6 cm and a standard deviation of 2.5 cm. (based on anthropométrie survey data from Gordon, Clauser, et al.). Consider the values of hip breadths of men that are unusual. What are the z scores that separate the unusual hip breadths from those that are usual? What are the hip breadths that separate the unusual hip breadths from those that are usual?

Problem 10CQQ For a sample of motorcycle speeds, name the values that constitute the 5-number summary.

Problem 13BSC Comparing Values. In Exercises, use z scores to compare the given values. Tallest Man and Woman As of this writing, the tallest living man is Sultan Kosen, who has a height of 247 cm. The tallest living woman is De-Fen Yao, who is 236 cm tall. Heights of men have a mean of 175 cm and a standard deviation of 7 cm. Heights of women have a mean of 162 cm and a standard deviation of 6 cm. Relative to the population of the same gender, who is taller? Explain.

Problem 10RE Mean or Median? A statistics class with 40 students consists of 30 students with no income, 10 students with small incomes from part-time jobs, and a professor with a very large income that is well deserved. Which is better for describing the income of a typical person in this class: mean or median? Explai

Problem 11BSC Usual and Unusual Values. In Exercises, consider a value to be unusual if its z score is less than ?2 or greater than 2. Earthquakes The Southern California Earthquake Data Center recorded magnitudes (Richter scale) of 10, 594 earthquakes in a recent year. The mean is 1.240 and the standard deviation is 0.578. Consider the magnitudes that are unusual. What are the magnitudes that separate the unusual earthquakes from those that are usual?

Usual and Unusual Values. In Exercises 9–12, consider a value to be unusual if its z score is less than ? 2 or greater than 2. Female Voice Based on data from Data Set 17 in Appendix B, the words spoken in a day by women have a mean of 16,215 words and a standard deviation of 7301 words. Consider the women with an unusual word count in a day. What are the numbers of words that separate the unusual word counts from those that are usual?

Comparing Values. In Exercises 13–16, use z scores to compare the given values. Oscars As of this writing, Sandra Bullock was the last woman to win an Oscar for Best Actress and Jeff Bridges was the last man to win for Best Actor. At the time of the awards ceremony, Sandra Bullock was 45 years of age and Jeff Bridges was 60 years of age. Based on Data Set 11 in Appendix B, the Best Actresses have a mean age of 35.9 years and a standard deviation of 11.1 years. The Best Actors have a mean age of 44.1 years and a standard deviation of 9.0 years. (All ages are determined at the time of the awards ceremony.) Relative to their genders, who was younger when winning the Oscar: Sandra Bullock or Jeff Bridges? Explain

Percentiles. In Exercises 17–20, use the following duration times (seconds) of 24 eruptions of the Old Faithful geyser in Yellowstone National Park. The duration times are sorted from lowest to highest. Find the percentile corresponding to the given time. 240 sec

Comparing Values. In Exercises 13–16, use z scores to compare the given values. Red Blood Cell Counts Based on Data Set 1 in Appendix B, males have red blood cell counts with a mean of 5.072 and a standard deviation of 0.395, while females have red blood cell counts with a mean of 4.577 and a standard deviation of 0.382. Who has the higher count relative to the sample from which it came: A male with a count of 4.91 or a female with a count of 4.32? Explain.

Percentiles. In Exercises 17–20, use the following duration times (seconds) of 24 eruptions of the Old Faithful geyser in Yellowstone National Park. The duration times are sorted from lowest to highest. Find the percentile corresponding to the given time. 213 sec

Percentiles. In Exercises 17–20, use the following duration times (seconds) of 24 eruptions of the Old Faithful geyser in Yellowstone National Park. The duration times are sorted from lowest to highest. Find the percentile corresponding to the given time. 250 sec

Percentiles. In Exercises 17–20, use the following duration times (seconds) of 24 eruptions of the Old Faithful geyser in Yellowstone National Park. The duration times are sorted from lowest to highest. Find the percentile corresponding to the given time. 260 sec

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. P 60

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. Q 3

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. P 40

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. P 75

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. P 50

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. P 25

In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. P 85

Boxplots. In Exercises 29–32, use the given data to construct a boxplot and identify the 5-number summary. Challenger Flights The following are the duration times (minutes) of all missions flown by the space shuttle Challenger.

Boxplots. In Exercises 29–32, use the given data to construct a boxplot and identify the 5-number summary. Old Faithful The following are the interval times (minutes) between eruptions of the Old Faithful geyser in Yellowstone National Park (based on data from the U.S. National Park Service).

Boxplots. In Exercises 29–32, use the given data to construct a boxplot and identify the 5-number summary. Speeds The following are speeds (mi/h) of cars measured with a radar gun on the New Jersey Turnpike (based on data from Statlib and authors Joseph Kadane and John Lamberth).

Boxplots. In Exercises 29–32, use the given data to construct a boxplot and identify the 5-number summary. Grooming Times The following are amounts of time (minutes) spent on hygiene and grooming in the morning by survey respondents (based on data from an SCA survey).

Boxplots from Larger Data Sets in Appendix B. In Exercises 33–36, use the given data sets from Appendix B. Pulse Rates Use the same scale to construct boxplots for the pulse rates of males and females from Data Set 1 in Appendix B. Use the boxplots to compare the two data sets.

Boxplots from Larger Data Sets in Appendix B. In Exercises 33–36, use the given data sets from Appendix B. Ages of Oscar Winners Use the same scale to construct boxplots for the ages of the best actresses and best actors from Data Set 11 in Appendix B. Use the boxplots to compare the two data sets.

Problem 35BSC Weights of Regular Coke and Diet Coke Use the same scale to construct boxplots for the weights of regular Coke and diet Coke from Data Set 19 in Appendix B. Use the boxplots to compare the two data sets.

Boxplots from Larger Data Sets in Appendix B. In Exercises 33–36, use the given data sets from Appendix B. Lead and IQ Use the same scale to construct boxplots for the full IQ scores (IQF) for the low lead level group and the high lead level group in Data Set 5 of Appendix B.

Outliers and Modified Boxplots Repeat Exercise 34 using modified boxplots. Identify any outliers as defined in Part 2 of this section. What do the modified boxplots show that the regular boxplots do not show?

Interpolation When finding percentiles using Figure 35, if the locator L is not a whole number, we round it up to the next larger whole number. An alternative to this procedure is to interpolate. For example, using interpolation with a locator of \(L=23.75\) leads to a value that is 0.75 (or 3/4) of the way between the 23rd and 24th values. Use this method of interpolation to find P 17 for the chocolate chip counts in Table 34. How does the result compare to the value that would be found by using Figure 35 without interpolation? Equation Transcription: Text Transcription: L=23.75

Statistical Literacy and Critical Thinking In Exercises 21–28, use the same list of 24 sorted Old Faithful eruption duration times given for Exercises 17–20. Find the indicated percentile or quartile. Q 1