statistics exam

Measures of Variability Statistics (exercises)

Aleksandra Pawłowska

May 12, 2020

Glossary

Range A measure of variability, defined to be the largest value minus the smallest value. Interquartile range (IQR) A measure of variability, defined to be the difference between the third and first quartiles. Variance A measure of variability based on the squared deviations of the data values about the mean. Standard deviation A measure of variability computed by taking the positive square root of the variance. Coefficient of variation A measure of relative variability computed by dividing the standard deviation by the mean and multiplying by 100.

Aleksandra Pawłowska Measures of Variability

Task 1

Consider a sample with data values of 10, 20, 12, 17, and 16. Com- pute the range and interquartile range.

Aleksandra Pawłowska Measures of Variability

Task 1 – solution

Consider a sample with data values of 10, 20, 12, 17, and 16. Com- pute the range and interquartile range. range = 10, IQR = 5

Aleksandra Pawłowska Measures of Variability

Task 2

Consider a sample with data values of 10, 20, 12, 17, and 16. Com- pute the variance and standard deviation.

Aleksandra Pawłowska Measures of Variability

Task 2 – solution

Consider a sample with data values of 10, 20, 12, 17, and 16. Com- pute the variance and standard deviation. s2 = 16, s = 4

Aleksandra Pawłowska Measures of Variability

Task 3

Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the range, interquartile range, variance, and standard deviation.

Aleksandra Pawłowska Measures of Variability

Task 3 – solution

Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the range, interquartile range, variance, and standard deviation. range = 19, IQR = 6.5, s2 = 34.57, s = 5.88.

Aleksandra Pawłowska Measures of Variability

Task 4

A bowler’s scores for six games were 182, 168, 184, 190, 170, and 174. Using these data as a sample, compute the following descriptive statistics:

1 Range 2 Variance 3 Standard deviation 4 Coefficient of variation

Aleksandra Pawłowska Measures of Variability

Task 4 – solution

1 Range 22 2 Variance 75.2 3 Standard deviation 8.67 4 Coefficient of variation 4.87%

Aleksandra Pawłowska Measures of Variability

Task 5 A home theater in a box is the easiest and cheapest way to provide surround sound for a home entertainment center. A sample of prices is shown here (Consumer Reports Buying Guide, 2004). The prices are for models with a DVD player and for models without a DVD player.

1 Compute the mean price for models with a DVD player and the mean price for models without a DVD player. What is the additional price paid to have a DVD player included in a home theater unit?

2 Compute the range, variance, and standard deviation for the two samples. What does this information tell you about the prices for models with and without a DVD player?

Aleksandra Pawłowska Measures of Variability

Task 5 – solution

1 Compute the mean price for models with a DVD player and the mean price for models without a DVD player. What is the additional price paid to have a DVD player included in a home theater unit? For models with DVD player x̄ = 410 and for models without DVD player x̄ = 310.

2 Compute the range, variance, and standard deviation for the two samples. What does this information tell you about the prices for models with and without a DVD player? For models with DVD player range = 200, s2 = 5500, s = 74.16 and for models without DVD player range = 70, s2 = 800, s = 28.28.

Aleksandra Pawłowska Measures of Variability

Task 6 Car rental rates per day for a sample of seven Eastern U.S. cities are as follows (The Wall Street Journal, January 16, 2004).

1 Compute the mean, variance, and standard deviation for the car rental rates.

2 A similar sample of seven Western U.S. cities showed a sample mean car rental rate of $38 per day. The variance and standard deviation were 12.3 and 3.5, respectively. Discuss any difference between the car rental rates in Eastern and Western U.S. cities.

Aleksandra Pawłowska Measures of Variability

Task 6 – solution

1 Compute the mean, variance, and standard deviation for the car rental rates. x̄ = 38, s2 = 97, s = 9.85.

2 A similar sample of seven Western U.S. cities showed a sample mean car rental rate of $38 per day. The variance and standard deviation were 12.3 and 3.5, respectively. Discuss any difference between the car rental rates in Eastern and Western U.S. cities. Eastern shows more variation.

Aleksandra Pawłowska Measures of Variability

Task 7

The Los Angeles Times regularly reports the air quality index for various areas of Southern California. A sample of air quality index values for Pomona provided the following data: 28, 42, 58, 48, 45, 55, 60, 49, and 50.

1 Compute the range and interquartile range. 2 Compute the sample variance and sample standard deviation. 3 A sample of air quality index readings for Anaheim provided a

sample mean of 48.5, a sample variance of 136, and a sample standard deviation of 11.66. What comparisons can you make between the air quality in Pomona and that in Anaheim on the basis of these descriptive statistics?

Aleksandra Pawłowska Measures of Variability

Task 7 – solution

1 Compute the range and interquartile range. range = 32, IQR = 10.

2 Compute the sample variance and sample standard deviation. s2 = 92.75, s = 9.63.

3 A sample of air quality index readings for Anaheim provided a sample mean of 48.5, a sample variance of 136, and a sample standard deviation of 11.66. What comparisons can you make between the air quality in Pomona and that in Anaheim on the basis of these descriptive statistics? Air quality in Anaheim is, on average, better than in Pomona, but also shows more variation.

Aleksandra Pawłowska Measures of Variability

Task 8 How do grocery costs compare across the country? Using a market basket of 10 items including meat, milk, bread, eggs, coffee, potatoes, cereal, and orange juice, Where to Retire magazine calculated the cost of the market basket in six cities and in six retirement areas across the country (Where to Retire, November/December 2003). The data with market basket cost to the nearest dollar are as follows:

1 Compute the mean, variance, and standard deviation for the sample of cities and the sample of retirement areas.

2 What observations can be made based on the two samples?

Aleksandra Pawłowska Measures of Variability

Task 8 – solution

1 Compute the mean, variance, and standard deviation for the sample of cities and the sample of retirement areas. For the sample of cities: x̄ = 33, s2 = 14.4, s = 3.79 and the sample of retirement areas: x̄ = 32, s2 = 3.6, s = 1.90.

2 What observations can be made based on the two samples?

Aleksandra Pawłowska Measures of Variability

Task 9

Scores turned in by an amateur golfer at the Bonita Fairways Golf Course in Bonita Springs, Florida, during 2005 and 2006 are as follows:

1 Use the mean and standard deviation to evaluate the golfer’s performance over the two-year period.

2 What is the primary difference in performance between 2005 and 2006? What improvement, if any, can be seen in the 2006 scores?

Aleksandra Pawłowska Measures of Variability

Task 9 – solution

1 Use the mean and standard deviation to evaluate the golfer’s performance over the two-year period. In 2005: x̄ = 76, s = 2.07 and in 2006: x̄ = 76, s = 5.26.

2 What is the primary difference in performance between 2005 and 2006? What improvement, if any, can be seen in the 2006 scores? There was, on average, no improvement in the scores in season 2006 comparing to the season 2005, but the scores in 2006 show more variation.

Aleksandra Pawłowska Measures of Variability