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Assignment 01 ASSIGNMENT 01 Unique Nr.: 798531 Fixed closing date: 13 MAY 2021 QUESTION 1 In testing whether the means of two populations are equal, we are given the summary statistics calculated from two independent samples as follows: 71 = 26 12 = 26 *1 = 7.30 $2 = 6.80 $1 = 1.05 $2 = 1.20 Assuming that the population variances are equal, which one of the following statements is correct? 1. The two independent samples are not normally distributed. 2. The pooled variance is equal to 0.3189. 3. The standard error of the sample mean difference in - x2 is equal to 1.2713. 4. The critical value for a two-tailed test at the 5% level of significance is 1.676. 5. The rejection region at 5% level of significance is / > 2.009. QUESTION 2 A company wants to compare the mean number of days of sick leave for two classes of employees, those with less or equal to five years of service versus those with more than five years of service. The sample sizes are nj = ny = 100 employees, the sample means are x1 = 241 and x2 = 240 and the standard deviations of the two populations are 1 = 8.2 days and oz = 5.7 days. The manager want to test the hypotheses: Ho : #1 /2. The p-value from the population difference in mean number of days of sick leave is equal to 1. 0.1587 2. 0.8413 3. 1.00 4. 0.3174 5. 0.9986QUESTION 3 Consider two samples of data drawn from two independent populations that are normally distrib uted. The summary statistics are given below *1 = 57 12 = 63 01 = 11.5 72 = 15.2 71 = 20 12 = 20 The analyst wants to test whether the population means differ at the 5% level of significance. Which one of the following statements is incorrect? 1. The hypotheses are: Ho : #1 - #2 =0 VS H1 : #1 - #2 #0. 2. The standard error of #1 - #2 is 4.262. 3. The test statistic is = -1.4078. 4. The rejection region is [z| > z = 1.96. 5. Conclusion: Ho is rejected at the 5% level of significance. QUESTION 4 Refer to the information given on question 3, the p-value is 1. 0.1586 2. 0.9207 3. 0.0808 4. 0.0793 5. 0.0193 QUESTION 5 Two random samples of 10 observations are drawn from two independent populations that are normally distributed. Consider the following summary statistics under the assumption that the two population variances are unknown but equal. X1 = 249 $2 = 272 51 = 35 $2 = 23 The 90% confidence interval for the difference between the two normal population means is 3 1. (-23 ; 38.1159) 2. (-57.1693 ; 11.1693) 3. (-45.9649 ; -0.0351) 4. (15.1159 ; 61.1159) 5. (-59.1519; 13.1519)QUESTION 6 A nutritionist wants to examine whether the average drink calories declined at cola-cola after the passage of the ordinance. The nutritionist obtains transaction data for 41 cola-cola cardholders and records their average drink calories prior to the ordinance and then after the ordinance. The nutritionist is informed that the two samples are normally distributed and the following summary statistics were obtained: The mean XD = 2.10 The sample variance s = 66.50 The test statistic = 1.63 The 95% confidence interval for the difference between the two samples is 1. (2.10 ; 2.5739) 2. (-0.4739; 4.6739) 3. (0.4739; 4.6739) 4. (0.0039; 4.8769) 5. (-0.4379; 4.9376) QUESTION 7 Refer to the information on question 6 and suppose a statistician wants to draw a conclusion based on the calculated results. Which one of the following statements is correct? 1. Fail to reject Ho since the statistic is higher than the critical value. 2. Reject Ho since the test statistic is smaller than the critical value. 3. Fail to reject Ho since the test statistic is smaller than the critical value. 4. Reject Ho since the test statistic is greater than the critical value. 5. Need more information. STA1502/011/0/2021 QUESTION 8 A production supervisor at a major chemical company must determine which of two-catalysts, cat- alysts A or catalysts B, maximizes the hourly yield of a chemical process. In order to compare the mean hourly yields obtained by using the two catalysts, the supervisor runs the process using each catalyst for five one-hour periods. It seems reasonable to regard the two catalysts as independent experiments with the population variances approximately equal. The results from the two samples catalysts (in rands per hour) are presented in the table below Catalyst A X1 = 811 S = 386 Catalyst B X2 = 750.2 S; = 484.2 nj = my = 5 The test statistic is 1. 435.1 2. 13.1924 3. 174.04 4. 46.0876 5. 4.6087 QUESTION 9 Two independent samples of sizes 25 and 35 are randomly selected from two normal populations with equal variances. Suppose the tutor wants to test the difference between the population means. The appropriate test statistic is 1. a Z-test with 60 degrees of freedom. 2. a standard normal variable. 3. approximately a sample normal distribution. 4. a student /-test with 58 degrees of freedom. 5. F-test with 24 and 34 degrees of freedom.QUESTION 10 In testing the hypotheses Ho : up = 5, os Hj : up > 5, two random samples selected from two normal populations produced the following summary statistics: np = 20, ip = 9, and sp = 7.5. A statistical test is performed to solve the problem at the 10% level of significance. Which of the following statements is incorrect? 5 1. Rejection region is z > to.10,19 = 1.328. 2. The test statistic is / = 3.5664. 3. Reject the null hypothesis at the 10% level of significance. 4. The lower confidence limit LCL = 6.1003 and the upper confidence limit UCL = 11.8997. 5. To apply the matched pair test, the two random samples selected have to be normally distrib uted QUESTION 11 Suppose the two random samples from two normal populations produced the following summary statistics: S- = 35 71 = 25 $4 = 70 n2 = 31 The 90% confidence interval for the ratio of the two population variances is 1. LCL = 0.2646 UCL = 0.945 2. LCL = 1.031 UCL = 3.88 3. LCL = 0.2466 UCL = 0.954 4. LCL = 0.126 UCL = 0.197 5. LCL = 0.2646 UCL = 0.97 QUESTION 12 After a recent AIDS awareness, the health department commissioned a market research com- pany to conduct a survey on its effectiveness. The aim was to establish whether the recall rate of teenagers differed from that young adults (20 - 30 years of age). The market research com- pany interviewed a random sample of 640 teenagers and 420 young adults. It was found that 362 teenagers and 260 young adults were able to recall the AIDS awareness. The management wants to test at the 5% level of significance, the hypothesis that there is an equal recall rate between teenagers and young adults. The test statistic for the difference between the two population proportions is 1. 0.5868 2. -1.7282 3. 0.0316 4. 1.6899 5. 1.9763 6 STA1502/011/0/2021 QUESTION 13 Consider the following information: 71 = 100, X1 = 60, n2 = 100, and X2 = 40. The statistics analyst wants to use the above information to test if there is a significant difference between the two population proportions. Which one of the following statements is correct? 1. The sample proportions are p1 = 0.40 and p2 = 0.60 2. The pooled proportion equals 0.05 3. The standard error for the difference between two population proportions is 0.7007 4. The test statistic for the difference between two population proportions is 2.2889 5. The 95% confidence interval estimate of (p1 - p2) is (0.0614 ; 0.3386)QUESTION 14 Consider the information given on question 13, the p-value for a one-tailed test is 1. 0.0023 2. 0.9977 3. 0.0046 4. 0.9974 5. 0.0026 QUESTION 15 A sample of size 100 candidates is selected from a population upon which there are 60 successful candidates successes, and another sample of size 150 is selected from a second population has 95 successful candidates. The standard error for the difference between the two population proportion is equal: 1. -0.5337 2. 0.7293 3. 0.0627 4. 0.2702 5. 0.0267 N QUESTION 16 A two-tailed test is conducted to test the difference between two population proportions. The two sample proportions are p1 = 0.20, p2 = 0.15, and the sample sizes are n1 = 160, n2 = 200 respectively. The upper limit for the 99% confidence interval for the difference P1 - P2 is 1. 0.04321 2. 0.1545 3. 0.103 4. 2.5800 5. 0.0532 QUESTION 17 A politician regularly polls her constituency to gauge her level of support among voters. This month, 652 out of 1158 voters support her. Suppose that the calculated test statistic is 6.5899. Can she conclude that support has increased by at least 10 percentage points? The hypotheses used for the test are Ho : P1 - P2 = 0.10 against HI : P1 - P2 > 0.10 Which one of the following statements is correct ? 1. Reject Ho at 5% level of significance. 2. Fail to reject Ho at the 5% level of significance. 3. Reject Ho at the 5% level and fail to reject Ho at 10% level of significance. 4. Fail to reject Ho at 2.5% level but reject Ho at 1% level of significance. 5. Reject Ho at 10% level and fail to reject Ho at 5% level of significance. QUESTION 18 Suppose the analyst wants to test whether two population variances are equal, this test will be upon the 1. difference between two sample variances. 2. ratio of the two sample variances. 3. difference between two population variances. 4. difference between two population means. 5. sample that follow a student- distribution with equal standard deviations.QUESTION 19 A statistician wants to test if two independent sample means are equal. Assume that the two populations are normally distributed in which people have enrolled in a diet program. The number of pounds recorded as lost at the completion of the program is as follows: Sample 1 7 9 6 15 7 10 8 12 $- = 9.071 X1 = 9.25 Sample 2 2 25 9 15 10 18 5 22 27 3 S; = 84.044 X2 = 13.6 The different calculations and conclusions of the statistician in his attempt to infer at the 5% level of significance that the population variances differ, are given below. Which one of the following statements is correct? 1. Ho : = 1 against #1 : " - 1 . 2. The test statistic is F = 0.0108. 3. The rejection region consists of either F F(0.975,7,9) -F(0.025,7,9) = 0.24. 4. The null hypothesis cannot be rejected. 5. At the 5% level of significance we can infer that the two population variances are unequal. QUESTION 20 Consider to the information given on question 17, the p-value is 1. 0.7257 2. 0.5279 3. 0.0000 4. 0.4721 5. 6.58 9 QUESTION 21 Suppose the analysis of variance (ANOVA) table is given as shown below Source of Degrees of Sum of Mean squares MS F variation freedom df squares SS SS7 5 SSE 54 16.2 Total 21.4 The critical value at 5% level of significance is 1. 2.40 2. 6.23 3. 2.28 4. 4.43 5. 2.89QUESTION 22 One - way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: 21 =7 X1 = 65 $1 = 4.2 12 = 8 X2 = 64 $2 = 4.9 13 =9 X3 = 61 $3 = 4.6 The mean square for error is 1. 110 2. 443.19 3. 125.23 4. 21.10 5. 21.1043 10 STA1502/011/0/2021 QUESTION 23 The following statistics were calculated based on samples drawn from three normal populations: Treatment Statistic 1 2 3 The size n 10 10 10 The sample mean 95 86 92 The sample standard deviation s |10 12 15 The manager make use the analysis of variance (ANOVA) table to determine whether there is a difference between the population means at the 5% level of significance. Which of the following statements is incorrect? 1. The grand mean of all the observation x = 91. 2. The sum of squares for treatment SST = 420. 3. The sum of squares for Error SSE = 4641. 4. The F statistic test = 1.343. 5. The F-critical value (one-tailed) = 3.34. QUESTION 24 If we apply Fisher's LSD procedure with a = 0.05 to determine which population means differ, given the following summary statistics: k =3 m1 = 10, n2 = 10, 13 = 10, MSE = 700, *1 = 128.7, #2 = 101.4, $3 = 133.7, the value of Fisher's Least Significant Difference (LSD) equals to 1. 700 2. 500 3. 140 4. 121.3985 5. 24.2797QUESTION 25 In one way ANOVA, suppose that there are five treatments with nj = ny = nj = 5 and my = ns =7. Then the mean square for error MSE equals SSE 1 . 4 SSE 2. 29 SSE 3. 24 SSE 4. 5 SSE 5. 12