Question

t) A national sporting good store wishes to use demographic information to predict its monthly sales, in $1000s. Thrity-eight, n=38, stores of the chain are randomly chosen across the country. It is known that each store is approximately the same size and carries the same merchandise. The geographic area from which a store draws its customers is known as the customer base. One of the variables is the percentage of the customer base who have graduated from high school.

(1 point) A national sporting good store wishes to use demographic information to predict its monthly sales, in $1000s. Thrity-eight, n 38, stores of the chain are randomly chosen across the country. It is known that each store is approximately the same size and carries the same merchandise. The geographic area from which a store draws its customers is known as the customer base. One of the variables is the percentage of the customer base who have graduated from high school. MonthlySales,-80 + β, PercentHSGrads, + ei MonthlySalesi-is the total sales in month i, in $1000s PercentHSGrads, is percentage of all customers in store i customer base that have graduated from high school A least-squares regression was ran in Minitab producing the following output: Scatter Plot: Sales to High School Graduates 4000 3500 3000 2500 2000 (webwork2 files/tmp/W2019STAT217/images/271fed85-8b2b-3430-aa98- 1000 500 50 S5 0 65 70 75 80 85 0 95 Percentage of Customer Base with a HS Dipoma aa322914205c 4d843331-a47a-38fb-8304-2e666ab0bcd4.png) Regression Analysis: MonthlySales versus PercentHSGrads Predictor Coef SD Constant 29701371 9.66 17.67 S 802.004 R-Sq- of DF SS Residual Error 23155564 Total 37 Using the partial Minitab output, answer the questions belovw (a) Estimate the model. Use two-decimals your estimation of the slope term, no decimals in the estimation of the y-intercept. Using the partial Minitab output, answer the questions belovw (a) Estimate the model. Use two-decimals your estimation of the slope term, no decimals in the estimation of the y-intercept. MonthlySales- 滋Percent HSGradsi (b) What percentage of the variation in a store's monthly sales cannot be explained by its linear dependency on the percentage of the customer base that are high school graduates? Enter your answer as a percentage, using two decimal places. (c) Does the data collected indicate that the monthly sales of a store can be expressed as a linear function the percentage of high school graduates in its customer base? Select the correct statisticaly hypotheses. (d) Using the F-test, test the statistical hypotheses determined in (c). Find the value of the test statistic, using two decimals in your answer cale (e) Testing the statistical hypotheses in (c) at α-0.05, you can conclude from this data that the be expressed as a linear function of the (f) Can you infer from this data that an increase of 1% to the percentage of high school graduates in the customer based will lead to an mean/average increase in the store's monthly sales by more than $50,0001? (i) Find the value of the test statistic, use two decimal places in your answer calc (i) Find the P-value of the result, using three decimals. P-value (g) A store located at a local mall has recently discovered that 90% of its customer base has a high school diploma. With 95% confidence, estimate this store's monthly sales for the current month. Note: You will need Σ381 Percent HSGrads,-2935.17 and Σ381 Percent HSGrads-228777 P-value (g) A store located at a local mall has recently discovered that 90% of its customer base has a high school diploma. with 95% confidence, estimate this store's monthly sales for the current month. Note: You will need Percent H SGrads, 2935.17andPercent HSGrads 228777 Lower Bound $1000s (use one decimal in your answer) Upper Bound $1000s (use one decimal in your answer) (h) A residual plot of the regression was consulted. Residual Plot Resporse is Sales in $1000s) 2000 500 1000- 500- (webwork2 files/tmp/W2019STAT217/images/271fed85-8b2b-3430-aa98- 500 1000 500 L000 2000 2500 Fitted Value aa322914205c 470a600d-0b58-33cf-bbae-c5001838bc73.png) What does this residual plot say about the condition(s) of the model? Pick the most appropriate answer. OA. The variation in the proportion of the customer based that are high school graduates is not the same for all stores. OB. The distribution in the monthly sales is Normally distributed. O C. The distribution in the monthly sales is not Normally distributed. O D. The variance in the monthly sales is not the same for all stores with different proportions of high school graduates in their respective customer base. O E. The variation in the proportion of the customer base that are high school graduates is the same for all stores. O F. The variance in the monthly sales is the same for all stores with different proportions of high school graduates in their respective customer base

Answer 1

$(a)~\widehat{MonthlySales}_i=-2970+59.66~PercentHSGrads_i\\ (b)~R^2=SSR/SST=7333350/(7333350+23155564)=0.2405\\ where,~SSR=sum~of~square~due~to~regression=7333350\\ SST=Total~sum~of~square=7333350+23155564=30488914\\ Answer~24.05\%.\\ (c)~Option~C.~H_0:\beta_1=0~H_A:\beta_1\neq 0.\\ (d)~F_{calc}=\left(\frac{\hat{\beta}_1}{SD(\hat{\beta}_1)} \right )^2=(59.66/17.67)^2=11.40\\ (e)~p-value=P(F>11.40|F\sim F_{1,36})=0.0018<0.05.\\ Hence~H_0~is~rejected.\\ Hence~we~can~conclude~that~the~monthly~sales~of~a~store~can~be~expressed~as~a~linear~function~\\ of~the~percentage~of~high~school~graduates~in~its~customer~base.\\ (f)~Null~hypothesis,~H_0:\beta_1=50~vs.~Alternative~hypothesis,~H_1:\beta_1>50\\ (i)~T_{cal}=\frac{\hat{\beta}_1-50}{SD(\hat{\beta}_1)}=(59.66-50)/17.67=0.55\\ (ii)~P-value=P(T>0.55|T\sim t_{36})= 0.293\\$

$Since~P-value>0.05~so~an~increase~of~1\%~to~the~percentage~of~high~school~graduates~in~the\\ ~customer~based~will~lead~to~an~average~increase~in~the~store's~monthly~sales~is~not~more~than~\$50000.\\ (g)~S_{xx}=228777-2935.17^2/38=2060.607,~t_{0.025,36}=2.0281\\ \bar{x}=2935.17/38=77.2413,~x_0=90,~y_0=-2970+59.66*90=2399.4,\\~S=802.004\\ Lower~bound=y_0-t_{0.025,36}\times S\times \sqrt{\left(\frac{1}{38}+\frac{(x_0-\bar{x})^2}{S_{xx}} \right )}=1871.6\\ Upper~bound=y_0+t_{0.025,36}\times S\times \sqrt{\left(\frac{1}{38}+\frac{(x_0-\bar{x})^2}{S_{xx}} \right )}=2927.2\\ (h)Option~D.$