Business Statistics (ECON 509)Homework # 4Problem # 1Maine Mountain Dairy claims that its eight-ounce low-fat yogurt cupscontain, on average, fewer calories than the eight-ounce low-fat yogurt cupsproduced by a competitor. A consumer agency wanted to check this claim. Asample of 27 such yogurt cups produced by this company showed that theycontained an average of 141 calories per cup. A sample of 25 such yogurtcups produced by its competitor showed that they contained an average of144 calories per cup. Assume that the two populations are normallydistributed with population standard deviations of 5.5 calories and 6.4calories, respectively.a. Make a 98% confidence interval for the difference between the meannumber of calories in the eight-ounce low-fat yogurt cups producedby the two companies.b. Test at the 1% significance level whether Maine Mountain DairyΒsclaim is true.c. Calculate the p-value for the test of part b. Based on this p-value,would you reject the null hypothesis if ? = .005? What if ? = .025?Problem # 2A car magazine is comparing the total repair costs incurred during the firstthree years on two sports cars, the T-999 and the XPY. Random samples of 45T-999s and 51 XPYs are taken. All 96 cars are three years old and havesimilar mileages. The mean of repair costs for the 45 T-999 cars is $3300 forthe first three years. For the 51 XPY cars, this mean is $3850. Assume that thestandard deviations for the two populations are $800 and $1000,respectively.a. Construct a 99% confidence interval for the difference between thetwo population means. Test the null hypothesis at the ? = 0.10 level ofsignificance.b. Using the 1% significance level, can you conclude that such meanrepair costs are different for these two types of cars?c. What would your decision be in part b if the probability of making aType I error were zero? Explain.Problem # 3The following information was obtained from two independent samplesselected from two normally distributed populations with unknown but equalstandard deviations.1 1 12 2 225 12 50 3 7520 14 60 3 10n , x . , s .n , x . , s .? ? ?? ? ?a. What is the point estimate of ?1 Β ?2?b. Construct a 95% confidence interval for ?1 Β ?2.c. Test at the 5% significance level if the two population means aredifferent.Problem # 4The management of a supermarket wanted to investigate whether the malecustomers spend less money on average than the female customers. A sampleof 35 male customers who shopped at this supermarket showed that theyspent an average of $80 with a standard deviation of $17.50. Another sampleof 40 female customers who shopped at the same supermarket showed thatthey spent an average of $96 w
ith a standard deviation of $14.40.a. Construct a 99% confidence interval for the difference between themean amounts spent by all male and all female customers at thissupermarket.b. Using the 2.5% significance level, can you conclude that the meanamount spent by all male customers at this supermarket is less thanthat spent by all female customers?Problem # 5An insurance company wants to know if the average speed at which mendrive cars is higher than that of women drivers. The company took a randomsample of 27 cars driven by men on a highway and found the mean speed tobe 72 miles per hour with a standard deviation of 2.2 miles per hour. Anothersample of 18 cars driven by women on the same highway gave a mean speedof 68 miles per hour with a standard deviation of 2.5 miles per hour. Assumethat the speeds at which all men and all women drive cars on this highwayare both normally distributed with unequal population standard deviations.a. Construct a 98% confidence interval for the difference between themean speeds of cars driven by all men and all women on this highway.b. Test at the 1% significance level whether the mean speed of cars drivenby all men drivers on this highway is higher than that of cars driven byall women drivers.Problem # 6A private agency claims that the crash course it offers significantly increasesthe writing speed of secretaries. The following table gives the scores of eightsecretaries before and after they attended this course. Before81758991657090After9772931107869115 a. Make a 90% confidence interval for the mean ?d of the populationpaired differences, where a paired difference is equal to the scorebefore attending the course minus the score after attending the courseb. Using the 5% significance level, can you conclude that attending thiscourse increases the writing speed of secretaries?Problem # 7Find the following confidence intervals for ?d assuming that the populationsof paired differences are normally distributed.a. n = 12, d = 17.5, sd = 6.3, confidence level = 99%.b. n = 27, d = 55.9, sd = 14.7, confidence level = 95%.Problem # 8A sample of 1000 observations taken from the first population gave x1 = 290.Another sample of 1200 observations taken from the second population gavex2 = 396.a. Find the point estimate of p1 Β p2.b. Make a 98% confidence interval for p1 Β p2.c. Will you reject the null hypothesis mentioned in part c at a significancelevel of 1%?Problem # 9Conduct the following tests of hypotheses assuming that the populations ofpaired differences are normally distributed.a. H0: ?d = 0, H1: ?d ? 0, n = 26, d = 9.6, sd = 3.9, ? = .05b. H0: ?d = 0, H1: ?d > 0, n = 15, d = 8.8, sd = 4.7, ? = .01