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The table below is used for the preliminary computations for finding the least squares line (the line of best fit) for the given pairs of x and y values. Xi Yi 2 Xi Xi Yi Yi 7 2 4 4 6 2 2 5 1 7 1 6 3 5 ∑Xi = ∑Yi = ∑ 2 Xi = ∑Xi Yi = ∑Yi = a. Complete the table b. Find SSxy (this is called sum of squares of cross product) c. Find XX SS (called, sum of squares of the X’s) d. Find YY SS (called, sum of squares of the Y’s) e. Find b, the slope of the regression line. f. Find a, the y-intercept of the regression line. g. the least squares line (the regression line, the line of best fit). h. calculate the sample correlation coefficient, r Where, ∑ ∑ ∑ = − n X Y SS X Y i i xy i i n X SS X i XX i ∑ = ∑ − 2 2 ( ) n Y SS Y i YY i 2 2 (∑ ) = ∑ − XX XY SS SS b = ; a = Y − bX n X X ∑ i = and n Y Y ∑ i = The least squares line is given by, Y = a + bX  The sample correlation coefficient is given, r is given by XX YY XY SS SS SS r = A marketing-research analyst is interested in examining the statement made by the makers that brand A cigarettes contain less than 3 milligrams of tar. The marketing-research analyst randomly selected 60 cigarettes and found the mean amount of tar to be 2.75 milligrams with a standard deviation of 1.5 milligrams. Do the data support the claim at the 5% significance level? Find the p-value. On the basis of the p-value, what is your conclusion? 2. An auditing firm was hired to determine if a particular defense plant was overstating the value of their inventory items. It was decided that 15 items would be randomly selected. For each item, the recorded amount, the audited exact) amount, and the difference between these two amounts (recorded - audited) were determined. Of particular interest was whether it could be demonstrated that the average difference exceeds $25, in which case the defense plant would be subject to a loss of contract and financial penalties. The following differences were obtained (in dollars): 17, 35, 31, 22, 50, 42, 56, 23, 27, 38, 20, 25, 43, 45, 21 a. Set up the appropriate hypotheses and test them using a significance level of α = .05. b. What is the p-value of the test? 3. In a discussion of SAT Mathematics (SATM) scores, someone comments: “Because only a minority of California high school students take the test, the scores overestimate the ability of typical high school seniors. I think that if all seniors took the test, the mean score would be no more than 450.” You decide to test this claim ( ) H0 and gave the SAT to a simple random sample of 500 seniors. Assume that the population standard deviation is 100. The test rejects H0 at the 1% level of significance. Is this test sufficiently sensitive to usually detect an increase of 10 points in the population mean score? 4. You want to see if a redesign of the cover of a mail-order catalog will increase sales. A very large number of customers will receive the original catalog, and a random sample of customers will receive the one with the new cover. For planning purposes, you are willing to assume that the sales from the new catalog will be approximately Normal with standard deviation of σ = 50 dollars and the mean for the original catalog will be µ = 25 dollars. You decide to use a sample of size 100. You wish to test H0 : µ = 25 Ha : µ > 25 a. Find the probability of a Type I error. b. Find the probability of a Type II error when µ = 28. c. Find the probability of a Type II error when µ = 30. 5. The supervisor of a group of assembly-line workers wanted to compare last year’s productivity (X) to this year’s productivity (Y) for each of the 20 employees that she supervise. In the past an approximate linear relationship has existed between these two variables. Last year the average productivity per worker was 9.5 items per hour. This year the average productivity per worker is 12.1 items per hour. The supervisor found the following sums for her 20 employees. 0.8 0.3 0.4 = = = Y Y X X XY SS SS SS a. Calculate the correlation coefficient. b. Calculate the least squares line. c. Calculate the sum of squares for error. 6. The owner of Grandmother’s Cake Shop would like to predict the quantity of cakes sold when they are marked at low prices. There are no restrictions on the quantity, because the shop can easily bake several cakes in an hour if the demand is stronger than predicted. Past data show the following results: NUMBER OF PRICE OF CAKE (X) CAKES SOLD (Y) 14 2.30 16 2.10 17 1.80 17 1.89 13 2.50 12 2.80 16 1.99 17 1.90 15 2.25 14 2.39 13 2.70 a Calculate the least squares line for X and Y 7. M&M plain chocolate candies come in six different colors: dark brown, yellow, red, orange green, and blue. According to the manufacturer, (Mars, Inc.), the color ration in each large production batch is 30% brown, 20% yellow, 20% red, 10% orange, 10% green, and 10% blue. To test this claim, a professor at Carleton College had students count the colors of M&M’s found in “fun size” bags of candy. The results for the 370 M&M’s are displayed in the table. Construct a test to determine whether the true percentages of the colors produced differ from the manufacturer’s stated percentages. Use α = .05

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null hypotheses : population mean is 3
alternative hypotheses: population mean is less than 3
n=60; sample mean m=2.75; sample standard deviation s=1.5; 5% level
df=60-1=59 t value= 1.67 (right-tail) 2 (two-tailed)...
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