Elementary Statistics 2nd Edition
Elementary Statistics 2nd Edition
2nd Edition
ISBN: 9781259724275
Author: William Navidi, Barry Monk
Publisher: McGraw-Hill Education
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Question
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Chapter 7.3, Problem 19E

(a)

To determine

To calculate: The population mean, μ and the population standard deviation, σ .

(a)

Expert Solution
Check Mark

Answer to Problem 19E

The population mean, μ is 75.4 and the population standard deviation, σ is 5.12 .

Explanation of Solution

Given information:The data is shown below in table.

    DateTemperature
    July 2169
    July 2275
    July 2379
    July 2483
    July 2471

Formula used:

  Population mean: μ=xNPopulation Standard Deviation: σ=σ2= (x μ ) 2 N

Calculation:

The population mean, μ and the population standard deviation, σ can be calculated by using above formulas as follows:

  Population mean: μ=xN=69+75+79+83+715Population mean: μ=75.4Population Standard Deviation: σ=σ2= (x μ ) 2 NPopulation Standard Deviation: σ= (6975.4) 2 + (7575.4) 2 +.....+ (7175.4) 2 5Population Standard Deviation: σ=5.12

(b)

To determine

To list: All samples of size, 2 drawn with replacement.

(b)

Expert Solution
Check Mark

Answer to Problem 19E

All 25 samples of size 2 (with replacement) are listed in Table 1 .

Explanation of Solution

Given information:The data is shown below in table.

    DateTemperature
    July 2169
    July 2275
    July 2379
    July 2483
    July 2471

Calculation:

There are 25 samples of size 2 with replacement as shown in Table 1 .

    Sample
    69,69
    69,75
    69,79
    69,83
    69,71
    75,69
    75,75
    75,79
    75,83
    75,71
    79,69
    79,75
    79,79
    79,83
    79,71
    83,69
    83,75
    83,79
    83,83
    83,71
    71,69
    71,75
    71,79
    71,83
    71,71

(c)

To determine

To compute:Sample mean of eachof the 25 samples, mean μX¯ of the sample means and the standard deviation, σX¯ of the sample means.

(c)

Expert Solution
Check Mark

Answer to Problem 19E

Sample means of eachof the 25 samples are shown in Table 2 .

The mean, μX¯ of the sample meansis 75.4 .

The standard deviation, σX¯ of the sample means is 3.6968 .

Explanation of Solution

Given information:The data is shown below in table.

    DateTemperature
    July 2169
    July 2275
    July 2379
    July 2483
    July 2471

Formula used:

  The mean of sampling distribution: μX¯= X ¯n     X¯ - sample mean      n - sample sizeThe standard deviation of sampling distribution: σX¯=σ X ¯  2= ( X ¯ μ X ¯ ) 2 n1       

Calculation:

The sample mean can be calculated by adding the two samples together and dividing by the sample size which is 2.

    SampleSample Mean, X¯
    69,6969+692=69
    69,7569+752=72
    69,7969+792=74
    69,8369+832=76
    69,7169+712=70
    75,6975+692=72
    75,7575+752=75
    75,7975+792=77
    75,8375+832=79
    75,7175+712=73
    79,6979+692=74
    79,7579+752=77
    79,7979+792=79
    79,8379+832=81
    79,7179+712=75
    83,6983+692=76
    83,7583+752=79
    83,7983+792=81
    83,8383+832=83
    83,7183+712=77
    71,6971+692=70
    71,7571+752=73
    71,7971+792=75
    71,8371+832=77
    71,7171+712=71

The mean, μX¯ of the sample means is the average of all sample means which is calculated by adding all 25 sample means and dividing by 25.

  The mean of the sample means: μX¯= X ¯n=69+72+74+........+75+77+7125The mean of the sample means: μX¯=75.4The standard deviation of the sample means: σX¯= (6975.4) 2 +.............+ (7175.4) 2 24The standard deviation of the sample means: σX¯=3.6968

(d)

To determine

To verify: The relationship between the population mean, μ and the mean of the sample means, μX¯

  (μX¯=μ) and the relationship between the population standard deviation, σ and the standard deviation of the sample means, σX¯

  (σX¯=σ2)

(d)

Expert Solution
Check Mark

Answer to Problem 19E

Based on the calculated values, the relationships of μX¯=μ(=75.4) and σX¯=σ2 can be verified as true.

Explanation of Solution

Given information:The data is shown below in table.

    DateTemperature
    July 2169
    July 2275
    July 2379
    July 2483
    July 2471

Calculation:

As per the calculations in part (a) and (c), the mean of the sample means, μX¯ and the population mean, μ are both equal to 75.4 . So, the relationship of μX¯=μ(=75.4) holds true.

Based on the calculations in part (a), the population standard deviation, σ is 5.12 and the standard deviation of the sample means, σX¯ is 3.6968 .

  σ2=5.122=3.62

  σ2 is approximately equal to σX¯ though there is a slight difference in second decimal place. But, approximately, the relationship of (σX¯=σ2) holds true.

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Chapter 7 Solutions

Elementary Statistics 2nd Edition

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