Probability and Statistics: Week 2 Exercise

In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set 13, 10, 6, 17, 11. (a) Use the defining formula, the computation formula, or a calculator to compute s. (Round your answer to one decimal place.) (b) Multiply each data value by 5 to obtain the new data set 65, 50, 30, 85, 55. Compute s. (Round your answer to one decimal place.)

(a)4.03 (b)20.18

One standard for admission to Redfield College is that the student must rank in the upper quartile of his or her graduating high school class. What is the minimal percentile rank of a successful applicant?

75

How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (°F) were taken from May to November in the vicinity of Furnace Creek. Compute the mean, median, and mode for these ground temperatures. (Enter your answers to one decimal place.) 148 154 168 174 184 181 182 184 181 168 181 165 154 146

Mean 169.28 Median 171 Mode 181

(b) Multiply each data value by 7. Compute the mode, median, and mean.

mode 42 median 49 mean 60.2

Consider the following numbers. 2 3 4 5 5 (a) Compute the mode, median, and mean. (b) If the numbers represented codes for the colors of T-shirts ordered from a catalog, which average(s) would make sense? (Select all that apply.) (c) If the numbers represented one-way mileages for trails to different lakes, which average(s) would make sense? (Select all that apply.) (d) Suppose the numbers represent survey responses from 1 to 5, with 1 = disagree strongly, 2 = disagree, 3 = agree, 4 = agree strongly, and 5 = agree very strongly. Which average(s) make sense? (Select all that apply.)

(a) Mean 5 Median 4 Mode 3.8 (b)mode (c)mode, median, mean (d)mode, median

Consider the mode, median, and mean. (a) Which average represents the middle value of a data distribution? (b) Which average represents the most frequent value of a data distribution? (c) Which average takes all the specific values into account?

(a) median (b) mode (c) mean

In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set 6, 6, 7, 10, 14. (a) Compute the mode, median, and mean. (Enter your answers to one decimal place.) (b) Add 5 to each of the data values. Compute the mode, median, and mean. (Enter your answers to one decimal place.) (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set?

(a) mode 6 median 7 mean 8.6 (b) mode 11 median 12 mean 13.6 (c)Adding the same constant c to each data value results in the mode, median, and mean increasing by c units.

Consider the following types of data that were obtained from a random sample of 49 credit card accounts. Identify all the averages (mean, median, or mode) that can be used to summarize the data. (Select all that apply.) (a) Outstanding balance on each account. (b) Name of credit card (e.g., MasterCard, Visa, American Express, etc.). (c) Dollar amount due on next payment.

(a) mode, median, mean (b) mode (c) mode, median, mean

Each of the following data sets has a mean of x = 10. (i) 8 9 10 11 12 (ii) 7 9 10 11 13 (iii) 7 8 10 12 13 (a) Without doing any computations, order the data sets according to increasing value of standard deviations. (b) Why do you expect the difference in standard deviations between data sets (i) and (ii) to be greater than the difference in standard deviations between data sets (ii) and (iii)? Hint: Consider how much the data in the respective sets differ from the mean.

(a)(i), (ii), (iii) (b) The data change between data sets (i) and (ii) increased the squared difference Σ(x - x)2 by more than data sets (ii) and (iii).

One indicator of an outlier is that an observation is more than 2.5 standard deviations from the mean. Consider the data value 80. (a) If a data set has mean 70 and standard deviation 5, is 80 a suspect outlier? (b) If a data set has mean 70 and standard deviation 3, is 80 a suspect outlier?

(a)No, since 80 is less than 2.5 standard deviations above the mean. (b)Yes, since 80 is more than 2.5 standard deviations above the mean.

Consider a data set with at least three data values. Suppose the highest value is increased by 10 and the lowest is decreased by 10. (a) Does the mean change? Explain. (b) Does the median change? Explain. (c) Is it possible for the mode to change? Explain.

(a)No, the sum of the data does not change. (b)No, changing the extreme data values does not affect the median. (c)Yes, depending on which data value occurs most frequently after the data are changed.

(b) Find the interquartile range.

5

In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. However, the lab score is worth 15% of your total grade, each major test is worth 24%, and the final exam is worth 37%. Compute the weighted average for the following scores: 72 on the lab, 88 on the first major test, 71 on the second major test, and 78 on the final exam.

77.82

Find the mean, median, and mode of the data set. 7 2 5 2 3

Mean 3.8 Median 3 Mode 2

Find the mean, median, and mode of the data set. 8 2 7 2 6 5

Mean 5 Median 5.5 Mode 2

The town of Butler, Nebraska, decided to give a teacher-competency exam and defined the passing scores to be those in the 70th percentile or higher. The raw test scores ranged from 0 to 100. Was a raw score of 82 necessarily a passing score? Explain.

No, it might have a percentile rank less than 70.

A personnel office is gathering data regarding working conditions. Employees are given a list of five conditions that they might want to see improved. They are asked to select the one item that is most critical to them. Which type of graph, circle graph or Pareto chart, would be the most useful for displaying the results of the survey? Why?

Pareto chart, because it shows the items in order of importance to employees.

What symbol is used for the standard deviation when it is a sample statistic? What symbol is used for the standard deviation when it is a population parameter?

Sample statistic: s. Population parameter: σ.

What is the relationship between the variance and the standard deviation for a sample data set?

The standard deviation is the square root of the variance.

Clayton and Timothy took different sections of Introduction to Economics. Each section had a different final exam. Timothy scored 83 out of 100 and had a percentile rank in his class of 72. Clayton scored 85 out of 100 but his percentile rank in his class was 70. Who performed better with respect to the rest of the students in the class, Clayton or Timothy? Explain your answer.

Timothy, since his percentile score is higher.

When computing the standard deviation, does it matter whether the data are sample data or data comprising the entire population? Explain.

Yes. The formula for s is divided by n − 1, while the formula for σ is divided by N.

(c) Illinois has a bachelor's degree percentage rate of about 26%. Into what quarter does this rate fall?

between the median and Q3

Which average - mean, median, or mode - is associated with the standard deviation?

mean

What percentage of the general U.S. population have bachelor's degrees? The Statistical Abstract of the United States, 120th Edition, gives the percentage of bachelor's degrees by state. For convenience, the data are sorted in increasing order. 17 18 18 18 19 20 20 20 21 21 21 21 22 22 22 22 22 22 23 23 24 24 24 24 24 24 24 24 25 26 26 26 26 26 26 27 27 27 27 27 28 28 29 31 31 32 32 34 35 38 (a) Select the box-and-whisker plot.

median 24 1st quartile 15

What symbol is used for the arithmetic mean when it is a sample statistic? What symbol is used when the arithmetic mean is a population parameter?

statistic, x; parameter, μ

Using the number of hospitals per state listed in the table above, make a stem-and-leaf display for the number of community hospitals per state. (Enter NONE in any unused answer blanks.)

0 8 1 1 2 5 6 9 2 1 7 7 3 5 7 8 4 1 2 7 5 1 2 3 9 6 1 6 8 7 1 8 8 9 0 2 6 8 10 1 2 7 11 3 3 7 9 12 2 3 9 13 3 3 6 14 8 16 2 17 5 19 3 20 9 22 7 23 1 6 42 1 44 0

Wetlands offer a diversity of benefits. They provide a habitat for wildlife, spawning grounds for U.S. commercial fish, and renewable timber resources. In the last 200 years, the United States has lost more than half its wetlands. Environmental Almanac gives the percentage of wetlands lost in each state in the last 200 years. For the 30 of the lower 48 states, the percentage loss of wetlands per state is as follows. 50 50 27 50 59 60 48 46 52 33 90 38 9 20 42 23 37 35 54 35 38 89 87 52 46 35 72 46 87 67 Make a stem-and-leaf display of these data. (Use the tens digit as the stem and the ones digit as the leaf. Enter NONE in any unused answer blanks.)

0 9 1 none 2 0 3 7 3 3 5 5 5 7 8 8 4 2 6 6 6 8 5 0 0 0 2 2 4 9 6 0 7 7 2 8 7 7 9 9 0

The American Medical Association Center for Health Policy Research included data, by state, on the number of community hospitals and the average patient stay (in days) in its publication State Health Care Data: Utilization, Spending, and Characteristics. The data (by state) are shown in the table.

Questions and answers below

At one hospital there is some concern about the high turnover of nurses. A survey was done to determine how long (in months) nurses had been in their current positions. The responses (in months) of 20 nurses were as follows. 25 4 7 16 27 38 29 44 14 10 9 25 31 28 30 13 22 33 10 38 Make a box-and-whisker plot of the data.

median 25 3rd quartile 31

Angela took a general aptitude test and scored in the 92nd percentile for aptitude in accounting. (a) What percentage of the scores were at or below her score? (b) What percentage were above?

(a) 92 (b) 8

Commercial dredging operations in ancient rivers occasionally uncover archaeological artifacts of great importance. One such artifact is Bronze Age spearheads recovered from ancient rivers in Ireland. A recent study gave the following information regarding discoveries of ancient bronze spearheads in Irish rivers. River Bann Blackwater Erne Shannon Barrow No. of spearheads 25 12 15 35 14

(a) Make a Pareto chart for these data. Shannon, Bann, Erne, Barrow, Blackwater, (b) Make a circle graph for these data Shannon 34.7% Blkwtr 11.9% Barr 13.9% Erne 14.9% Bann 24.8%

How old are professional football players? The 11th Edition of The Pro Football Encyclopedia gave the following information. Random sample of pro football player ages in years: 24 23 25 23 30 29 28 26 33 29 24 37 25 23 22 27 28 25 31 29 25 22 31 29 22 28 27 26 23 21 25 21 25 24 22 26 25 32 26 29 (a) Compute the mean, median, and mode of the ages. (Enter your answers to one decimal place.) (b) Compare the averages. Does one seem to represent the age of the pro football players most accurately? Explain.

(a) mean 26.25 median 22.5 mode 25 (b)The averages are very close. The median seems to represent the ages most accurately.

Consider the data set. 2, 3, 4, 5, 6 (a) Find the range. (b) Use the defining formula to compute the sample standard deviation s. (Round your answer to two decimal places.) (c) Use the defining formula to compute the population standard deviation σ. (Round your answer to two decimal places.)

(a)4 (b)1.58 (c)1.41

Your friend is thinking about buying shares of stock in a company. You have been tracking the closing prices of the stock shares for the past 90 trading days. Which type of graph for the data, histogram or time-series, would be best to show your friend? Why?

A time-series graph because the pattern of stock prices over time is more relevant than the frequency of a range of closing prices.

In this problem, we explore the effect on the standard deviation of adding the same constant to each data value in a data set. Consider the following data set. 9, 15, 14, 14, 5 (a) Use the defining formula, the computation formula, or a calculator to compute s. (Enter your answer to one decimal place.) (b) Add 3 to each data value to get the new data set 12, 18, 17, 17, 8. Compute s. (Enter your answer to one decimal place.) (c) Compare the results of parts (a) and (b). In general, how do you think the standard deviation of a data set changes if the same constant is added to each data value?

(a)4.27 (b)4.27 (c)Adding the same constant c to each data value results in the standard deviation remaining the same.

(c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant c? (d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s = 3.7 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations?

(c)20.18 (d)No