1.57 Potassium from potatoes. Refer to Exercise 1.30 (page 24) where you examined the potassium absorption of a group of 27 adults who ate a controlled diet that included 40 mEq of potassium from potatoes for five days.

1.58 Potassium from a supplement. Refer to Exercise 1.31 (page 24) where you examined the potassium absorption of a group of 29 adults who ate a controlled diet that included 40 mEq of potassium from a supplement for five days.

1.59 Potassium from potatoes. Refer to Exercise 1.30 (page 24) where you examined the potassium absorption of a group of 27 adults who ate a controlled diet that included 40 mEq of potassium from potatoes for five days.

1.60 Potassium from a supplement. Refer to Exercise 1.31 (page 24) where you examined the potassium absorption of a group of 29 adults who ate a controlled diet that included 40 mEq of potassium from a supplement for five days.

1.61 Potassium from potatoes. Refer to Exercise 1.30 (page 24) where you examined the potassium absorption of a group of 27 adults who ate a controlled diet that included 40 mEq of potassium from potatoes for five days. In Exercise 1.30, you used a stemplot to examine the distribution of the potassium absorption.

1.62 Potassium from a supplement. Refer to Exercise 1.31 (page 24) where you examined the potassium absorption of a group of 29 adults who ate a controlled diet that included 40 mEq of potassium from a supplement for five days. In Exercise 1.31, you used a stemplot to examine the distribution of the potassium absorption.

1.63 Compare the potatoes with the supplement. Refer to Exercises 1.30 and 1.31 (page 24). Use a back-to-back stemplot to display the data for the two sources of potassium. Use the stemplot to compare the two distributions and write a short summary of your findings.

1.64 Potassium sources. Refer to Exercises 1.30 and 1.31 (page 24). Use side-by-side boxplots in to describe the distributions.

1.65 Gosset’s data on double stout sales. William Sealy Gosset worked at the Guinness Brewery in Dublin and made substantial contributions to the practice of statistics.²³ In his work at the brewery, he collected and analyzed a great deal of data. Archives with Gosset’s handwritten tables, graphs, and notes have been preserved at the Guinness Storehouse in Dublin.²⁴ In one study, Gosset examined the change in the double stout market before and after World War I (1914–1918). For various regions in England and Scotland, he calculated the ratio of sales in 1925, after the war, as a percent of sales in 1913, before the war. Here are the data:

1.66 Measures of spread for the double stout data. Refer to the previous exercise.

1.67 Are there outliers in the double stout data? Refer to the previous two exercises.

1.68 Smolts. Smolts are young salmon at a stage when their skin becomes covered with silvery scales and they start to migrate from freshwater to the sea. The reflectance of a light shined on a smolt’s skin is a measure of the smolt’s readiness for the migration. Here are the reflectances, in percents, for a sample of 50 smolts:²⁵

1.69 Measures of spread for smolts. Refer to the previous exercise.

1.70 Are there outliers in the smolt data? Refer to the previous two exercises.

1.71 Potatoes. A quality product is one that is consistent and has very little variability in its characteristics. Controlling variability can be more difficult with agricultural products than with those that are manufactured. The following table gives the weights, in ounces, of the 25 potatoes sold in a 10-pound bag:

1.72 The alcohol content of beer. Brewing beer involves a variety of steps that can affect the alcohol content. A website gives the percent alcohol for 159 domestic brands of beer.²⁶

1.73 Outlier for alcohol content of beer. Refer to the previous exercise.

1.74 Calories in beer. Refer to the previous two exercises. The data set also lists calories per 12 ounces of beverage.

1.75 Median versus mean for net worth. A report on the assets of American households says that the median net worth of U.S. families is $81,200. The mean net worth of these families is $534,600.²⁷ What explains the difference between these two measures of center?

1.76 Create a data set. Create a data set with seven observations for which the median would change by a large amount if the smallest observation were deleted.

1.77 Mean versus median. A small accounting firm pays each of its seven clerks $55,000, three junior accountants $80,000 each, and the firm’s owner $650,000. What is the mean salary paid at this firm? How many of the employees earn less than the mean? What is the median salary?

1.78 Be careful about how you treat the zeros. In computing the median income of any group, some federal agencies omit all members of the group who had no income. Give an example to show that the reported median income of a group can go down even though the group becomes economically better off. Is this also true of the mean income?

1.79 How does the median change? The firm in Exercise 1.77 gives no raises to the clerks and junior accountants, while the owner’s take increases to $500,000. How does this change affect the mean? How does it affect the median?

1.80 Metabolic rates. Calculate the mean and standard deviation of the metabolic rates in Example 1.32 (page 38), showing each step in detail. First find the mean $\overline{x}$ by summing the seven observations and dividing by 7. Then find each of the deviations $x_i-\overline{x}$ and their squares. Check that the deviations have sum 0. Calculate the variance as an average of the squared deviations (remember to divide by n − 1). Finally, obtain s as the square root of the variance.

1.81 Earthquakes. Each year there are about 900,000 earthquakes of magnitude 2.5 or less that are usually not felt. In contrast, there are about 10 of magnitude 7.0 that cause serious damage.²⁸ Explain why the average magnitude of earthquakes is not a good measure of their impact.

1.82 IQ scores. Many standard statistical methods that you will study in Part II of this book are intended for use with distributions that are symmetric and have no outliers. These methods start with the mean and standard deviation, $\overline{x}$ and s. For example, standard methods would typically be used for the IQ and GPA data in Table 1.3 (page 26).

1.83 Mean and median for two observations. The Mean and Median applet allows you to place observations on a line and see their mean and median visually. Place two observations on the line by clicking below it. Why does only one arrow appear?

1.84 Mean and median for three observations. In the Mean and Median applet, place four observations on the line by clicking below it, three close together near the center of the line and one somewhat to the right of these two.

1.85 Mean and median for seven observations. Place seven observations on the line in the Mean and Median applet by clicking below it.

1.86 Imputation. Various problems with data collection can cause some observations to be missing. Suppose a data set has 20 cases. Here are the values of the variable x for 10 of these cases:

The values for the other 10 cases are missing. One way to deal with missing data is called imputation. The basic idea is that missing values are replaced, or imputed, with values that are based on an analysis of the data that are not missing. For a data set with a single variable, the usual choice of a value for imputation is the mean of the values that are not missing. The mean for this data set is 15.

1.87 A standard deviation contest. This is a standard deviation contest. You must choose four numbers from the whole numbers 10 to 20, with repeats allowed.

1.88 Longleaf pine trees. The Wade Tract in Thomas County, Georgia, is an old-growth forest of longleaf pine trees (Pinus palustris) that has survived in a relatively undisturbed state since before the settlement of the area by Europeans. A study collected data on 584 of these trees.²⁹ One of the variables measured was the diameter at breast height (DBH). This is the diameter of the tree at 4.5 feet and the units are centimeters (cm). Only trees with DBH greater than 1.5 cm were sampled. Here are the diameters of a random sample of 40 of these trees:

1.89 Weight gain. A study of diet and weight gain deliberately overfed 15 volunteers for eight weeks. The mean increase in fat was $\overline{x}=2.41$ kilograms, and the standard deviation was $s=1.25$ kilograms. What are $\overline{x}$ and s in pounds? (A kilogram is 2.2 pounds.)

1.90 Changing units from inches to centimeters. Changing the unit of length from inches to centimeters multiplies each length by 2.54 because there are 2.54 centimeters in an inch. This change of units multiplies our usual measures of spread by 2.54. This is true of IQR and the standard deviation. What happens to the variance when we change units in this way?

1.91 A different type of mean. The trimmed mean is a measure of center that is more resistant than the mean but uses more of the available information than the median. To compute the 10% trimmed mean, discard the highest 10% and the lowest 10% of the observations and compute the mean of the remaining 80%. Trimming eliminates the effect of a small number of outliers. Compute the 10% trimmed mean of the service time data in Table 1.2 (page 17). Then compute the 20% trimmed mean. Compare the values of these measures with the median and the ordinary untrimmed mean.

1.92 Changing units from centimeters to inches. Refer to Exercise 1.88 (page 50). Change the measurements from centimeters to inches by multiplying each value by 0.39. Answer the questions from that exercise and explain the effect of the transformation on these data.