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Sagot :
Testing the hypothesis, we have that:
a)
The null hypothesis is: [tex]H_0: \mu_1 - \mu_2 = 0[/tex]
The alternative hypothesis is: [tex]H_1: \mu_1 - \mu_2 \neq 0[/tex]
b)
The p-value of the test is of 0, which is less than 0.05, thus, the conclusion is that the population mean customer satisfaction scores for the two retailers are different.
c)
The 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers is (0.8775, 1.1225). The interval is entirely positive, which means that the first retailer appears to have the greater customer satisfaction.
Item a:
At the null hypothesis, we test if they have the same satisfaction score, that is:
[tex]H_0: \mu_1 - \mu_2 = 0[/tex]
At the alternative hypothesis, we test if they have different scores, that is:
[tex]H_1: \mu_1 - \mu_2 \neq 0[/tex]
Item b:
First, we find the test statistic, given by:
[tex]z = \frac{X - \mu}{s}[/tex]
For this problem, X is the difference of sample means, thus:
[tex]X = x_1 - x_2 = 86 - 85 = 1[/tex]
[tex]\mu[/tex] is the value tested at the null hypothesis, that is, [tex]\mu = 0[/tex].
The standard error is:
[tex]s = \sqrt{(\frac{\sigma_1}{n_1})^2 + (\frac{\sigma_2}{n_2})^2}[/tex]
For this problem, [tex]\sigma_1 = \sigma_2 = 12, n_1 = 250, n_2 = 300[/tex]
Then
[tex]s = \sqrt{(\frac{12}{250})^2 + (\frac{12}{300})^2} = 0.0625[/tex]
[tex]z = \frac{X - \mu}{s}[/tex]
[tex]z = \frac{1 - 0}{0.0625}[/tex]
[tex]z = 16[/tex]
Two-tailed test, so the p-value is P((|z| > 16), which is 2 multiplied by the p-value of z = -16.
z = -16 has a p-value of 0, thus 2 x 0 = 0.
The p-value of the test is of 0, which is less than 0.05, thus, the conclusion is that the population mean customer satisfaction scores for the two retailers are different.
Item c:
The confidence interval is:
[tex]X \pm zs[/tex]
z is the critical value, which is z with a p-value of [tex]\frac{1 + \alpha}{2}[/tex], in which [tex]\alpha[/tex] is the confidence level.
In this problem, 95% confidence interval, thus [tex]\alpha = 0.95[/tex], and z has a p-value of [tex]\frac{1 + 0.95}{2} = 0.975[/tex], so z = 1.96.
Then
[tex]X - zs = 1 - 1.96(0.0625) = 0.8775[/tex]
[tex]X + zs = 1 + 1.96(0.0625) = 1.1225[/tex]
The 95% confidence interval for the difference between the population mean customer satisfaction scores for the two retailers is (0.8775, 1.1225). The interval is entirely positive, which means that the first retailer appears to have the greater customer satisfaction.
A similar problem is given at https://brainly.com/question/23777908
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