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: Observe the given table carefully. x f 3 1 4 1 5 0 6 2 7 5 8 3 9 3 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 = 3 and f = 1 , the product x ⋅ f = 3 ⋅ 1 = 3. For x = 4 and f = 1 , the product x ⋅ f = 4 ⋅ 1 = 4. For x = 5 and f = 0 , the product x ⋅ f = 5 ⋅ 0 = 0. For x = 6 and f = 2 , the product x ⋅ f = 6 ⋅ 2 = 12. For x = 7 and f = 5 , the product x ⋅ f = 7 ⋅ 5 = 35. For x = 8 and f = 3 , the product x ⋅ f = 8 ⋅ 3 = 24. For x = 9 and f = 3 , the product x ⋅ f = 9 ⋅ 3 = 27. The sum of these products is 3 + 4 + 0 + 12 + 35 + 24 + 27 = 105. Therefore, ∑ ( x ⋅ f ) = 105. The sum of the frequencies is 1 + 1 + 0 + 2 + 5 + 3 + 3 = 15. Therefore, ∑ f = 15. The mean is then ∑ ( x ⋅ f ) ∑ f = 105 15 = 7

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ISBN: 9780134434681

Author: Tom Pirnot

Publisher: PEARSON

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Chapter 14.3, Problem 15E

To determine

To find:

The mean and standard deviation for the following table

x f

3 1

4 1

5 0

6 2

7 5

8 3

9 3

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Chapter 14.3, Problem 14E

Chapter 14.3, Problem 16E

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>tt 1:32 > trend.1m 1m (sales > summary(trend.1m) - tt) #3###23 (i) #### Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2107.220 57.997 36.332e-16 *** tt -43.500 3.067 -14.18 7.72e-15 *** > trend = ts (fitted (trend.1m), start-start (sales), freq-frequency (sales)) sales trend ###23%23 (ii) #### as.numeric((1:32 %% 4) > X > q1 > q2 > q3 > 94 = = = = - as.numeric((1:32 %% 4) as.numeric((1:32 %% 4) as.numeric((1:32 %% 4) == 1) 2) == == 3) == 0) > season.lm = 1m (resid (trend.1m) 0+q1 + q2 + q3 + q4) #3##23%23 (iii) #### > summary(season.1m) Coefficients: Estimate Std. Error t value Pr(>|t|) q1 -38.41 43.27 -0.888 0.38232 92 18.80 43.27 0.435 0.66719 q3 -134.78 43.27 -3.115 0.00422 ** 94 154.38 43.27 3.568 0.00132 ** > season = ts (fitted (season.lm), start=start (sales), freq=frequency (sales)) > Y X season %23%23%23%23 (iv) #### >ar (Y, aic=FALSE, order.max=1) #23%23%23%23 (v) #### Coefficients: 1 0.5704 Order selected 1 sigma 2 estimated as 9431 > ar(Y, aic=FALSE,…

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Chapter 14 Solutions

Mathematics All Around (6th Edition)

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The mean and standard deviation for the following table x f 3 1 4 1 5 0 6 2 7 5 8 3 9 3

To find: The mean and standard deviation for the following table x f 3 1 4 1 5 0 6 2 7 5 8 3 9 3

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Chapter 14 Solutions

To find:

The mean and standard deviation for the following table

x f

3 1

4 1

5 0

6 2

7 5

8 3

9 3