In order to estimate the difference between the average hourly wages of employees of two branches of a department store, the following data

Question

In order to estimate the difference between the average hourly wages of employees of two branches of a department store, the following data have been gathered. Downtown Store North Mall Store Sample size 25 20 Sample mean $9 $8 Sample standard deviation $2 $1 Refer to Exhibit 10-7. A 95% interval estimate for the difference between the two population means is

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Mackenzie 2 weeks 2021-10-08T08:00:08+00:00 1 Answer 0

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    2021-10-08T08:01:25+00:00

    Answer:

     (9-8) -2.02 \sqrt{\frac{2^2}{25} +\frac{1^2}{20}}= 0.0743

     (9-8) +2.02 \sqrt{\frac{2^2}{25} +\frac{1^2}{20}}= 1.926

    And we are 9% confidence that the true mean for the difference of the population means is given by:

     0.0743 \leq \mu_1 -\mu_2 \leq 1.926

    Step-by-step explanation:

    For this problem we have the following data given:

    \bar X_1 = 9 represent the sample mean for one of the departments

    \bar X_2 = 8 represent the sample mean for the other department

    n_1 = 25 represent the sample size for the first group

    n_2 = 20 represent the sample size for the second group

    s_1 = 2 represent the deviation for the first group

    s_2 =1 represent the deviation for the second group

    Confidence interval

    The confidence interval for the difference in the true means is given by:

     (\bar X_1 -\bar X_2) \pm t_{\alpha/2} \sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}

    The confidence given is 95% or 9.5, then the significance level is \alpha=0.05 and \alpha/2 =0.025. The degrees of freedom are given by:

     df=n_1 +n_2 -2= 20+25-2= 43

    And the critical value for this case is  t_{\alpha/2}=2.02

    And replacing we got:

     (9-8) -2.02 \sqrt{\frac{2^2}{25} +\frac{1^2}{20}}= 0.0743

     (9-8) +2.02 \sqrt{\frac{2^2}{25} +\frac{1^2}{20}}= 1.926

    And we are 9% confidence that the true mean for the difference of the population means is given by:

     0.0743 \leq \mu_1 -\mu_2 \leq 1.926

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