Calculate the sample standard deviation and sample variance for the following frequency distribution of hourly wages for a sample of pharmac

Question

Calculate the sample standard deviation and sample variance for the following frequency distribution of hourly wages for a sample of pharmacy assistants.

Class Frequency
8.26 20
10.01-11.75 38
11.76 36
13.51-15.25 25
15.26-17.00 27

Sample Variance: ___________
Sample Standard Deviation: _________

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Jezebel 3 years 2021-08-23T04:13:45+00:00 1 Answers 18 views 0

Answers ( )

  1. thienhuong
    0
    2021-08-23T04:14:52+00:00
    Reply

    Answer:

    (a) The sample variance is 16.51

    (a) The sample standard deviation is 4.06

    Step-by-step explanation:

    Given

    \begin{array}{cc}{Class} & {Frequency} & 8.26 - 10.00 & 20 &10.01-11.75 & 38 &11.76 - 13.50& 36 & 13.51-15.25 &25&15.26-17.00 &27 &\ \end{array}

    Solving (a); The sample variance.

    First, calculate the class midpoints.

    This is the mean of the intervals.

    i.e.

    x_1 = \frac{8.26+10.00}{2} = \frac{18.26}{2} = 9.13

    x_2 = \frac{10.01+11.75}{2} = \frac{21.76}{2} = 10.88

    x_3 = \frac{11.76+13.50}{2} = \frac{25.26}{2} = 12.63

    x_4 = \frac{13.51+15.25}{2} = \frac{28.76}{2} = 14.38

    x_5 = \frac{15.26+17.00}{2} = \frac{32.26}{2} = 16.13

    So, the table becomes:

    \begin{array}{ccc}{Class} & {Frequency} & {x} & 8.26 - 10.00 & 20&9.13 &10.01-11.75 & 38 &10.88&11.76 - 13.50& 36 &12.63& 13.51-15.25 &25&14.38&15.26-17.00 &27 &16.13\ \end{array}

    Next, calculate the mean

    \bar x = \frac{\sum fx}{\sum f}

    \bar x = \frac{20*9.13 + 38 * 10.88+36*12.63+25*14.38+27*16.13}{20+38+36+25+27}

    \bar x = \frac{1845.73}{146}

    \bar x = 12.64

    Next, the sample variance is:

    \sigma^2 = \frac{\sum f(x - \bar x)^2}{\sum f - 1}

    So, we have:

    \sigma^2 = \frac{20*(9.13-12.63)^2 + 38 * (10.88-12.63)^2 +...........+27 * (16.13 -12.63)^2}{20+38+36+25+27-1}

    \sigma^2 = \frac{2393.6875}{145}

    \sigma^2 = 16.51

    The sample standard deviation is:

    \sigma = \sqrt{\sigma^2}

    \sigma = \sqrt{16.51}

    \sigma = 4.06

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