Explanation Definition: 1. While computing the mean of a frequency distribution, write all products x ⋅ f of the values and their frequencies in a new column of the table. Represent the sum of these products by ∑ ( x ⋅ f ) . Denote the sum of the frequencies by ∑ f . The mean is then ∑ ( x ⋅ f ) ∑ f . 2. If x is a data value in a set whose mean is x ¯ , then x − x ¯ is called x’s deviation from the mean. 3. We denote the standard deviation of a sample of n data values by s = ∑ ( x − x ¯ ) 2 n − 1 . 4. We calculate the standard deviation, s, of a sample that is given as a frequency distribution as s = ∑ ( x − x ¯ ) 2 ⋅ f n − 1 . Here f is the frequency of data value x , and n = ∑ f is the number of data values in the distribution. Calculation: From the given table understand the frequency of each data value. x f 4.24 1 8.00 1 8.25 1 6.00 1 5.53 1 4.90 1 3.07 1 4.25 1 The frequency of each data value is 1 because the given table lists the data for single person from each state. First find the mean of the given data set. To compute the mean, let us find all products x ⋅ f of the given values and their respective frequencies. For x = 4.24 and f = 1 , the product x ⋅ f = 4.24 ⋅ 1 = 4.24. For x = 8.00 and f = 1 , the product x ⋅ f = 8.00 ⋅ 1 = 8.00. For x = 8.25 and f = 1 , the product x ⋅ f = 8.25 ⋅ 1 = 8.25. For x = 6.00 and f = 1 , the product x ⋅ f = 6.00 ⋅ 1 = 6.00. For x = 5.53 and f = 1 , the product x ⋅ f = 5.53 ⋅ 1 = 5.53. For x = 4.90 and f = 1 , the product x ⋅ f = 4.90 ⋅ 1 = 4.90. For x = 3.07 and f = 1 , the product x ⋅ f = 3.07 ⋅ 1 = 3.07. For x = 4.25 and f = 1 , the product x ⋅ f = 4.25 ⋅ 1 = 4.25. The sum of these products is 4.24 + 8.00 + 8.25 + 6.00 + 5.53 + 4.90 + 3.07 + 4.25 = 44.24. Therefore, ∑ ( x ⋅ f ) = 44.24. The sum of the frequencies is 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 To determine b. To explain: Whether the answer in part a) represents the average state income tax paid by the people of given states

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ISBN: 9780134434681

Author: Tom Pirnot

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Question

Chapter 14.3, Problem 21E

To determine

a.

To find:

The mean and standard deviation of the distribution given by the following table:

State Income tax (%)

Arizona 4.24

California 8.00

Hawaii 8.25

Georgia 6.00

New Jersey 5.53

New Mexico 4.90

Pennsylvania 3.07

Michigan 4.25

To determine

b.

To explain:

Whether the answer in part a) represents the average state income tax paid by the people of given states

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Chapter 14.3, Problem 20E

Chapter 14.3, Problem 22E

Chapter 14 Solutions

Mathematics All Around (6th Edition)

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