Problem 1: A supermarket chain is redesigning the checkout lanes and is considering two designs

Problem 1: A supermarket chain is redesigning the checkout lanes and is considering two designs. The following summary shows information on customer checkout times at two stores where the two new systems have been installed: System A System B Sample size ni = 22 n2 = 25 sample mean 21 = 4.1 minutes 12 = 3.4 minutes Sample std. dev. $1 = 2.2 minutes S2 = 1.5 minutes Let Mi be the true population mean checkout time of lanes with System A and likewise up the true population mean of lanes with System B. Management is interested in discerning whether the mean checkout times of the two systems differ. A) (2 pts) Formulate the null and alternative hypotheses. B) (5 pts] State the appropriate test statistic and its distribution. C) [8 pts) Compute the test statistic. D) [5 pts) At a 5% significance level, state your conclusion in the context of this study. [NOTE: Use 40 degrees of freedom] Problem 2: A researcher wants to study the difference between the percentages of users of two toothpastes who will never switch to another toothpaste. In a sample of 500 users of Toothpaste A, 100 said that they will never switch to another toothpaste. In another sample of 400 users of Toothpaste B, 68 said that they will never switch to another toothpaste. Let pi be the true population proportion of users of Toothpaste A who will never switch brands, and likewise p2 be the true population proportion of users of Toothpaste B. A) (2 pts] What is the point estimate for p? - P2? B) [8 pts) Construct the 99% confidence interval for p? - P2. C) (4 pts) Interpret and analyze the interval computed in (B). D) Perform a hypothesis test to test the claim: Proportion of users of Toothpaste A who will never switch to another toothpaste is unequal to than the proportion of users of Toothpaste B who will never switch. i) [2 pts) Formulate the null and alternative hypotheses. ii) [5 pts] Compute the test statistic. iii) (4 pts) At the 1% level of significance, state your conclusion in the context of this study. Problem 5: The following table provides information on prices of digital cameras and number of megapixels of the cameras. Price Megapixels 180 8 200 10 230 7 120 8 470 15 140 8 180 10 310 12 250 10 110 7 Regression analysis is conducted to predict price (Y) of the digital camera based on the megapixels (X). Price 100 150 200 250 300 350 400 450 1 8 10 12 14 Megapixels A) [5 pts) What does the scatter diagram indicate about the association between the two variables? B) To approximate the linear association between Price and Megapixels, the following output was obtained from R using simple linear regression analysis: Call: Im(formula = price ~ mp, data = my_data) Residuals: Min 1Q Median 3Q Max -58.16 -34.00 -9.00 16.82 106.80 Coefficients: Estimate Std. Error t value Pr(t) (Intercept) -145.027 66.320 -2.187 0.060223 mp 38.319 6.772 5.658 0.000477 *** Signif. codes: O ****' 0.001 ***' 0.01 '*' 0.05 ·.' 0.1 ''1 Residual standard error: 50.9 on 8 degrees of freedom Multiple R-squared: 0.8001, Adjusted R-squared: 0.7751 F-statistic: 32.01 on 1 and 8 DF, p-value: 0.0004769 Based on the above output, is there a significant linear association between the price of a camera and the megapixels? i) (2 pts] Answer the above at the 5% level of significance. ii) [4 pts) Write the estimated regression equation. iii) (7 pts) Is this equation a good fit for the data? Why or why not? iv) (7 pts] Using the equation in (ii), predict the expected price of a digital camera when it has 9 megapixels.