<p> DEPARTMENT OF MATHEMATICS</p><p>ASSIGNMENT 2 (15%)</p><p>Instructions:</p><p>1. This assignment is an individual work. </p><p>2. For each question that involves calculation, please use the correct formula and show the steps involved clearly.</p><p>3. Submission date: Week 14 </p><p>SEM 2 2015/2016 MTH 3003 (PJJ) 1. A random sample of 130 human body temperatures had a mean of 98.25 0F and a standard deviation of 0.73 0F. (a) Find the point estimate for the average of human body temperature and calculate its margin of error. (b) Construct a 99% confidence interval for the average of human body temperature. Interpret this interval. (c) Using 5% significance level, test whether the average of human body temperature is different from 98 0F. </p><p>2. A new process for producing synthetic diamonds can be operated at a profitable level only if the average weight of the diamonds is greater than 0.5 karat. To evaluate the profitability of the process, six diamonds are generated with recorded weights (in karat) are as follows: 0.46 0.61 0.52 0.48 0.57 0.54 (a) Construct a 95% confidence interval for the average weight of the diamonds produced by the process. Interpret this interval. (b) Do the six measurements present sufficient evidence to indicate that the average weight of the diamonds produced by the process is in excess of 0.5 karat?.</p><p>3. The mean hotel room rates in the US were reported to be $85.69 and $84.58 per day in years 2000 and 2001 respectively. Suppose that these mean rates were based on random samples of 1000 hotel rooms for year 2000 and 1100 hotel rooms for year 2001. Assume that the standard deviation of the rates for the two samples were $18.50 and $18 respectively.</p><p>(a) Let m1 and m2 be the mean rates for all hotel rooms in the US in years 2000 and</p><p>2001 respectively. What are the point estimate of m1- m 2 and its margin of error?. </p><p>(b) Find a 90% confidence interval for the difference between m1 and m2 . Interpret this interval. (c) Using 1% significance level, can you conclude that the mean rates for all hotel rooms in the US in year 2000 is higher than the mean rates for all hotel rooms in the US in year 2001?. </p><p>SEM 2 2015/2016 MTH 3003 (PJJ) 4. Two independent random samples of sizes n1 = 4 and n2 = 5 are selected from each of two normal populations with equal variances. </p><p>Population 1 12 3 8 5 Population 2 14 7 7 9 6 (a) Calculate s2 , the pooled estimator of s 2 . (b) Find a 99% confidence interval for the difference between the two population means. Interpret this interval. (c) Suppose you wish to detect the difference between the population means. State the null and alternative hypotheses for the test. (d) Find the rejection region for the test in part c for a = 0.01 and conduct the test.</p><p>5. In a study to compare the effects of two pain relievers it was found that of n1 = 200 randomly selected individuals who used the first pain reliever, 93% indicated that it</p><p> relieved their pain. Of n2 = 450 randomly selected individuals who used the second pain reliever, 96% indicated that it relieved their pain. (a) Find a 99% confidence interval for the difference in the proportions experiencing relief from pain for these two pain relievers. (b) Based on the confidence interval in part a, is there sufficient evidence to indicate a difference in the proportion experiencing relief for the two pain relievers?. Explain. (c) Test at 1% significance level whether there is a difference in the proportion experiencing relief for the two pain relievers. State your conclusion. (d) Do the conclusion in part b support your conclusion in part c?.</p><p>6. Regardless of age, about 20% of American adults participate in fitness activities at least twice a week. However, these fitness activities change as the people get older, and occasionally participants become nonparticipants as they age. In a local survey of n =100 adults over 40 years old, a total of 15 people indicated that they participated in a fitness activity at least twice a week. Do these data indicate that the participation rate for adults over 40 years of age is significantly less than the 20%. Test using 5% significance level.</p><p>7. A paired-difference experiment was conducted to compare the means of two populations: Pairs Population 1 2 3 4 5 1 1.3 1.6 1.1 1.4 1.7 2 1.2 1.5 1.1 1.2 1.8</p><p>SEM 2 2015/2016 MTH 3003 (PJJ) (a) Do the data provide sufficient evidence to indicate that m1 differs from m2 ?. Test using a = 0.05.</p><p>(b) Find a 95% confidence interval for (m1- m 2 ) . Compare your interpretation of the confidence interval with your test results in part a.</p><p>8. A random sample of size n = 7 from a normal population produced these measurements: 1.4 3.6 1.7 2.0 3.3 2.8 2.9 (a) Calculate the sample variance, s2 . (b) Construct a 95% confidence interval for the population variance, s 2 .</p><p>2 2 (c) Test H0 :s = 0.8 versus Ha :s 0.8 using a = 0.05. State your conclusion.</p><p>9. The variability in the amount of impurities present in a batch of chemical used for a particular process depends on the length of time the process is in operation. A manufacturer using two production lines, 1 and 2, has made a slight adjustment to line 2, hoping to reduce the variability as well as the average amount of impurities in the</p><p> chemical. Samples of n1 = 25 and n2 = 25 measurements from the two batches yield these means and variances: </p><p>2 x1=3.2 s 1 = 1.04 2 x2=3.0 s 2 = 0.51 (a) Suppose the manufacturer wish to test whether the process variability is less for line 2. State the null and alternative hypotheses for the test. (b) What is the appropriate test statistic to be used if the manufacturer wish to test whether the process variability is less for line 2?. (c) Do the data present sufficient evidence to indicate that the process variability is less for line 2?. Test using a = 0.05.</p><p>SEM 2 2015/2016 MTH 3003 (PJJ)</p>