Question 1

What measure of central location uses all of the observations in its calculation? _____________

Accepted Answer

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Question 2

A sample of the personnel files of eight male employees revealed that, during a six-month period, they lost the following number of days due to illness: 2, 0, 6, 3, 10, 4, 1 and 2. What is the mean deviation (in days)?

Accepted Answer

First, we need to find the mean of the given data, which is:

mean = (2+0+6+3+10+4+1+2) / 8
     = 28 / 8
     = 3.5

Next, we need to find the deviation of each data point from the mean:

2 - 3.5 = -1.5
0 - 3.5 = -3.5
6 - 3.5 = 2.5
3 - 3.5 = -0.5
10 - 3.5 = 6.5
4 - 3.5 = 0.5
1 - 3.5 = -2.5
2 - 3.5 = -1.5

To find the mean deviation, we need to add up the absolute values of these deviations and divide by the total number of data points:

mean deviation = (1.5 + 3.5 + 2.5 + 0.5 + 6.5 + 0.5 + 2.5 + 1.5) / 8
              = 18 / 8
              = 2.25

Therefore, the closest choice is D, 2 3/8.

Question 3

The distribution of a sample of the outside diameters of PVC gas pipes approximates a symmetrical, bell-shaped distribution. The arithmetic mean is 14.0 inches, and the standard deviation is 0.1 inches. About 68 percent of the outside diameters lie between what two amounts?

Accepted Answer

According to the empirical rule, approximately 68% of the data falls within one standard deviation from the mean for any symmetrical, bell-shaped distribution. In this case, one standard deviation is 0.1 inches, so we can find the range of outside diameters that are within one standard deviation of the mean by adding and subtracting 0.1 inches from the mean:
Mean - 1 standard deviation = 14.0 - 0.1 = 13.9 inches
Mean + 1 standard deviation = 14.0 + 0.1 = 14.1 inches
Therefore, about 68% of the outside diameters lie between 13.9 and 14.1 inches, which corresponds to choice C.

Question 4

Based on the Empirical Rule, what percent of the observations will lie between plus or minus two standard deviations from the mean?

Accepted Answer

The Empirical Rule states that approximately 68% of observations will fall within one standard deviation of the mean, 95% within two standard deviations of the mean, and 99.7% within three standard deviations of the mean. Therefore, the answer is A) 95%.

Question 5

Production of passenger cars in Japan increased from 3.94 million in 1999 to 6.74 million in 2009. What is the geometric mean annual percent increase?

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Question 6

In a frequency distribution, what measure of central location is used for the class with the largest number of observations? ______________

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Question 7

A sample of assistant professors on the business faculty at state supported institutions in Ohio revealed the mean income to be $72,000 for 9 months with a standard deviation of $3,000. Using Chebyshev's Theorem, what proportion of the faculty earns more than $66,000 but less than $78,000?

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Question 8

A sample of the monthly amounts spent for food by families of four receiving food stamps approximates a symmetrical distribution. The sample mean is $150 and the standard deviation is $20. Using the Empirical Rule, about 95 percent of the monthly food expenditures are between what two amounts?

Accepted Answer

The Empirical Rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations of the mean, and 99.7% falls within three standard deviations of the mean. 
In this case, since the sample approximates a symmetrical distribution, we can use the Empirical Rule to estimate that about 95% of the monthly food expenditures are between one standard deviation below and one standard deviation above the mean. 
One standard deviation below the mean is $150 - $20 = $130, and one standard deviation above the mean is $150 + $20 = $170.
Therefore, about 95% of the monthly food expenditures are between $130 and $170, and the best choice is D, which includes this range.

Question 9

A population consists of all the weights of all defensive tackles on Sociable University's football team. They are: Johnson, 204 pounds; Patrick, 215 pounds; Junior, 207 pounds; Kendron, 212 pounds; Nicko, 214 pounds; and Cochran, 208 pounds. What is the population standard deviation (in pounds)?

Accepted Answer

The population standard deviation is calculated by taking the square root of the average of the squared deviations from the mean. The mean weight is 210 pounds, and the squared deviations are 36, 25, 9, 4, 16, and 4. The average of these squared deviations is 15.67, and the square root of 15.67 is approximately 4.

Question 10

The U.S. Federal Aviation Administration reported that passenger revenues on international flights increased from $528 million in 1986 to $5,100 million in 2009. What is the geometric mean annual percent increase in international passenger revenues?

Accepted Answer

The geometric mean annual percent increase can be calculated using the formula: [(Final Value/Initial Value)^(1/Number of Years)] - 1. Plugging in the values: [(5100/528)^(1/23)] - 1 = 0.104 or 10.4%.

Question 11

If the sample variance for a frequency distribution consisting of hourly wages was computed to be 10, what is the sample standard deviation?

Accepted Answer

The sample standard deviation is the square root of the sample variance. Therefore, taking the square root of 10 gives us approximately 3.16. Therefore, the correct answer is C: $3.16.

Question 12

The number of students at a local university increased from 2,500 students to 5,000 students in 10 years. Based on a geometric mean, the university grew at an average percentage rate of

Accepted Answer

To calculate the average percentage rate of growth, we use the geometric mean formula:

Geometric mean = (final value / initial value)^(1/number of years)

Plugging in the values, we get:

Geometric mean = (5000/2500)^(1/10) 
Geometric mean = 1.071

Therefore, the university grew at an average percentage rate of 7.1% per year. The correct answer is C.

Question 13

The weights (in grams) of the contents of several small bottles are 4, 2, 5, 4, 5, 2 and 6. What is the sample variance?

Accepted Answer

The sample variance is calculated by finding the mean of the data, then finding the squared differences from the mean, summing them, and dividing by the number of data points minus one. The mean is 4, the squared differences are 0, 4, 1, 0, 1, 4, 4, summing to 14. Dividing by 6 gives a variance of 2.33.

Question 14

Each person who applies for an assembly job at Robert's Electronics is given a mechanical aptitude test. One part of the test involves assembling a plug-in unit based on numbered instructions. A sample of the length of time it took 42 persons to assemble the unit was organized into the following frequency distribution.  What is the standard deviation (in minutes)?

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Question 15

The net annual sales of a sample of small retail clothing stores were organized into the following relative frequency distribution.  What is the mean net sales (in $millions)?

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Question 16

The ages of all the patients in the isolation ward of the hospital are 38, 26, 13, 41 and 22. What is the population variance?

Accepted Answer

The population variance is calculated by taking the mean of the squared differences from the mean. Here, the mean age = (38 + 26 + 13 + 41 + 22) / 5 = 140 / 5 = 28. The squared differences from the mean are (38-28)^2, (26-28)^2, (13-28)^2, (41-28)^2, and (22-28)^2, which are 100, 4, 225, 169, and 36 respectively. The mean of these squared differences is (100 + 4 + 225 + 169 + 36) / 5 = 534 / 5 = 106.8.

Question 17

Samples of the wires coming off the production line were tested for tensile strength. The statistical results (in PSI) were:  According to the Empirical Rule, the middle 95 percent of the wires tested between approximately what two values?

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Question 18

Which measure of central location is used to determine an average annual percent increase?

Accepted Answer

The geometric mean is used to calculate an average annual percent increase because it takes into account the compounding effect of growth over time. The arithmetic mean would not be appropriate because it assumes a linear relationship between the data points, which may not be the case for percent increase over time. The weighted mean and mode are not commonly used for calculating average percent increase.

Question 19

In the calculation of the arithmetic mean for grouped data, which value is used to represent all the values in a particular class?

Accepted Answer

The class midpoint is the most suitable value to represent all the values in a particular class in the calculation of the arithmetic mean for grouped data. This is because it is the center point of the class interval and provides a more accurate representation of the values within the class than the upper or lower limits. Using the frequency or cumulative frequency would only give information about the number of data points in the class and not the actual values themselves.

Question 20

The Investment Research Institute reported in its Mutual Fund Fact Book that the number of mutual funds increased from 5725 in 1999 to 7977 in 2009. What is the geometric mean annual percent increase in the number of funds?

Accepted Answer

The geometric mean annual percent increase is calculated using the formula: [(Final Value/Initial Value)^(1/Number of Years)] - 1. Here, it is [(7977/5725)^(1/10)] - 1, which equals approximately 0.0337 or 3.37%.

Quiz 3: Describing Data: Numerical Measures