BackUsing Chebychev’s Theorem Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 100, the mean interval between Old Faithful’s eruptions is 101.56 minutes and the standard deviation is 42.69 minutes. Using Chebychev’s Theorem, determine at least how many of the intervals lasted between 16.18 minutes and 186.94 minutes. (Adapted from Geyser Times)
Finding the Sample Mean and Standard Deviation for Grouped Data In Exercises 39 and 40, make a frequency distribution for the data. Then use the table to find the sample mean and the sample standard deviation of the data set.
3 3 5 3 8 0 3 9 6 6 7 1 6 3 2 6 9 1 8 5 0 2 3 4 9
5 8 1 9 7 6 9 6 7 0 6 3 8 6 8 7 3 8 9 3 7 2 4 4 1
In Exercises 25 and 26, find the range, mean, variance, and standard deviation of the population data set.
The mileages (in thousands of miles) for a rental car company’s fleet.
4 2 9 12 15 3 6 8 1 4 14 12 3 3
Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.
n = 6
x̄ = 7
s ≈ 2
In Exercises 27 and 28, find the range, mean, variance, and standard deviation of the sample data set.
Salaries (in dollars) of a random sample of teachers
62,222 56,719 50,259 45,120 47,692 45,985 53,489 71,534
The mean sale per customer for 40 customers at a gas station is $32.00, with a standard deviation of $4.00. Using Chebychev’s Theorem, determine at least how many of the customers spent between $24.00 and $40.00.
From a random sample of airplanes, the number of defects found in their fuselages are listed. Find the sample mean and the sample standard deviation of the data.
Refer to the sample statistics from Exercise 5 and determine whether any of the house prices below are unusual. Explain your reasoning.
a. $225,000
Refer to the sample statistics from Exercise 5 and determine whether any of the house prices below are unusual. Explain your reasoning.
d. $147,000
Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.
Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.
a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.
Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.
The speeds for eight vehicles are listed. Using the sample statistics from Exercise 29, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning.
70, 78, 62, 71, 65, 76, 82, 64
Use frequency distribution formulas to estimate the sample mean and the sample standard deviation of the data set in Exercise 2.
Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.
40 35 49 53 38 39 40
37 49 34 38 43 47 35
c. Each employee in the sample takes a pay cut of $2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.
Mean Absolute Deviation Another useful measure of variation for a data set is the mean absolute deviation (MAD). It is calculated by the formula
MAD = Σ |x − x̄| / n.
b. Find the mean absolute deviation of the data set in Exercise 16. Compare your result with the sample standard deviation obtained in Exercise 16.
Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.
42 36 48 51 39 39 42
36 48 33 39 42 45 50
b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.