Chapter 3 (homework)

When a distribution is mound-shaped symmetrical, what is the general relationship among the values of the mean, median, and mode?

The mean, median, and mode are approximately equal.

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.

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

(b) 5 (c) between the median and Q3

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. 8, 6, 14, 6, 14 (a) Use the defining formula, the computation formula, or a calculator to compute s. (Enter your answer to four decimal places.) (b) Add 8 to each data value to get the new data set 16, 14, 22, 14, 22. Compute s. (Enter your answer to four decimal places.) (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) 4.0988 (b) 4.0988 (c) Adding the same constant c to each data value results in the standard deviation remaining the same.

Consider the data set. 2, 3, 6, 7, 8 (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) 6 (b) 2.59 (c) 2.32

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

Consider the following ordered data. 6 9 9 10 11 11 12 13 14 (a) Find the low, Q1, median, Q3, and high. (b) Find the interquartile range. (c) Make a box-and-whisker plot

(a) Low: 6 Q1: 9 median: 11 Q3: 12.5 high: 14 (b) 3.5

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, the sum of the data does not change. (c) Yes, depending on which data value occurs most frequently after the data are changed.

Consider a data set of 15 distinct measurements with mean A and median B. (a) If the highest number were increased, what would be the effect on the median and mean? Explain. (b) If the highest number were decreased to a value still larger than B, what would be the effect on the median and mean? (c) If the highest number were decreased to a value smaller than B, what would be the effect on the median and mean?

(a) The mean would increase while the median would remain the same. (b) The mean would decrease while the median would remain the same. (c) Both the mean and median would decrease

Consumer Reports rated automobile insurance companies and gave annual premiums for top-rated companies in several states. The figure below shows box plots for annual premiums for urban customers (married couple with one 17-year-old son) in three states. The box plots were all drawn using the same scale on a TI-84Plus/TI-83Plus calculator. (a) Which state has the lowest premium? the highest? (b) Which state has the highest median premium? (c) Which state has the smallest range of premiums? the smallest interquartile range? (d) The other set of figures give the five-number summaries generated on the TI-84Plus/TI-83Plus calculators for the box plots. Match the five-number summaries to the appropriate box plots.

(a) lowest, California; highest, Pennsylvania (b) Pennsylvania (c) smallest range, California; smallest IQR, Texas

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 7 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: 13 median: 14 mean: 15.6 (c) Adding the same constant c to each data value results in the mode, median, and mean increasing by cunits.

Some data sets include values so high or so low that they seem to stand apart from the rest of the data. These data are called outliers. Outliers may represent data collection errors, data entry errors, or simply valid but unusual data values. It is important to identify outliers in the data set and examine the outliers carefully to determine if they are in error. One way to detect outliers is to use a box-and-whisker plot. Data values that fall beyond the limits Lower limit: Q1 − 1.5 ✕ (IQR)Upper limit: Q3 + 1.5 ✕ (IQR) where IQR is the interquartile range, are suspected outliers. In the computer software package Minitab, values beyond these limits are plotted with asterisks (*). Students from a statistics class were asked to record their heights in inches. The heights (as recorded) were as follows. 65 72 68 64 60 55 73 71 52 63 61 74 69 67 74 50 4 75 67 62 66 80 64 65 (a) Make a box-and-whisker plot of the data. (b) Find the value of the interquartile range (IQR). (c) Multiply the IQR by 1.5 and find the lower and upper limits. (Enter your answers to one decimal place.) lower limit upper limit (d) Are there any data values below the lower limit? Above the upper limit? List any suspected outliers. What might be some explanations for the outliers?

(b) 10 (c) Lower limit= 46.5 Upper limit= 86.5 (d) Yes, 4 is below the lower limit and is probably an error.

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

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

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.

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

mean

Find the mean, median, and mode of the data set. 9 2 5 2 4

mean: 4.4 median: 4 mode: 2

Find the mean, median, and mode of the data set. 9 4 8 4 6 5

mean: 6 median: 5.5 mode: 4

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