General Instruction: Answer the problems given below. Show your complete solution and illustrations (if there is). 1. Suppose the following data are selected randomly from a population of normally distributed values with unknown variance. 40 39 45 45 49 41 47 44 56 53 39 42 44 38 57 58 a. Compute the mean and standard deviation. b. Construct a 90% confidence interval to estimate the population mean u. c. Find the lower and upper confidence limits. d. Interpret the interval. 2. Independent random samples were selected from two populations with unknown and unequal variances. Population 1 Population 2 1 = 18 n- = 12 X = 8.15 T = 6.15 s, 2 = 4.2 $ 2 = 2.4 Construct a 90% confidence interval to estimate the difference in the population means #1 - Hz. 3. A study is conducted to compare the performance of students with more than one personal electronic gadget and those with only one. A number of them were taken as subjects for the study. The mean grades of these students and these standard deviations are shown below. Students Mean Standard deviation Sample size With one gadget X1 = 83 $ = 12 n, =7 With more than one gadget *, = 79 $2 = 13 n, = 5 Is it possible to conclude that there is no significant difference in the mean grades of the two types of students at a 95% confidence level?4. A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven for 30,000 km revealed a sample mean of 0.12 cm or tread remaining with a standard deviation of 0.035 cm.. a. Construct a 99% confidence interval for the population mean amount of tread remaining for tires of this manufacturer. b. Would it be reasonable to conclude that after 30,000 km, the population mean amount of tread remaining is 0.11 cm. 5. A school administrator would like to investigate the performance of faculty in two departments based on the evaluation and feedback of the students. Assuming the sampling error is to be limited to no more than 8% and two sample sizes should be the same, how many students must be randomly selected for evaluations for each department? Use a 90% confidence level. 6. A researcher wishes to estimate the 95% confidence interval of the proportion of students who show interest in webcasting and E-learning. A previous record shows that 30% of the students are interested. What is the minimum sample size necessary if the researcher aims for a margin of error of 30%? 7. The following observations are a sample from a normal population. 56 63 68 69 73 61 65 72 73 75 62 70 71 78 77 66 71 76 81 83 67 74 79 79 88 a. Calculate the mean and standard deviations. b. Construct a 90% confidence interval to estimate the population mean a. c. Find the lower and upper confidence limits. d. Interpret the interval