Answer :
Answer:
1. The critical value to be used for constructing the confidence interval is 1.415
2. The standard error of the sampling distribution to be used in constructing the confidence interval, is approximately 6.260
Step-by-step explanation:
The number of mail-order receipts, n₁ = 8 sales receipts
The mean amount of sale, [tex]\bar x_1[/tex] = $69.30
The standard deviation of the sample, σ₁ = $26.25
The number of internet sales receipts, n₂ = 17 sales receipts
The mean amount of sale, [tex]\bar x_2[/tex] = $75.90
The standard deviation of the sample, σ₂ = $28.25
1. Where the population variance are not equal, we have;
[tex]\left (\bar{x}_1-\bar{x}_{2} \right ) - t_{c}\sqrt{\dfrac{\sigma _{1}^{2}}{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}< \mu _{1}-\mu _{2}< \left (\bar{x}_1-\bar{x}_{2} \right ) + t_{c}\sqrt{\dfrac{\sigma _{1}^{2}}{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}[/tex]
Where;
[tex]t_c[/tex] = The critical value for constructing the confidence interval
[tex]\sqrt{\dfrac{\sigma _{1}^{2}}{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}[/tex] = [tex]S.E_{\bar{x}_1-\bar{x}_{2}} =[/tex] The standard error for constructing the confidence interval
The degrees of freedom, df = n₁ - 1 = 8 - 1 = 7
The t-value at 90% confidence level, and degrees of freedom of 7, [tex]t_{90\%, 7}[/tex], from the tables is given as follows;
[tex]t_c[/tex] = [tex]t_{90\%, \, 7}[/tex] = 1.415
The critical value for constructing the confidence interval = [tex]t_c[/tex] = 1.415
2. The standard error of the sampling distribution to be used in constructing the confidence interval, [tex]S.E._{\bar{x}_1-\bar{x}_{2}}[/tex], is given as follows
[tex]S.E_{\bar{x}_1-\bar{x}_{2}} = \sqrt{\dfrac{\sigma _{1}^{2}}{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}[/tex]
[tex]S.E._{\bar{x}_1-\bar{x}_{2}} = \sqrt{\dfrac{26.25^{2}}{8}-\dfrac{28.25^{2}}{17}} \approx 6.260[/tex]
[tex]S.E._{\bar{x}_1-\bar{x}_{2}}[/tex] ≈ 6.260
The standard error of the sampling distribution to be used in constructing the confidence interval, [tex]S.E._{\bar{x}_1-\bar{x}_{2}}[/tex] ≈ 6.260