(a) If all the observations in a data set are identical, then the variance for this data set iszero.(b) If P(A) = 0.4 and P(B) = 0.5, then P(A AND B) = 0.2.(c) The mean is always equal to the median for a normal distribution.(d) A 95% confidence interval is wider than a 90% confidence interval of the same parameter.(e) In a two-tailed hypothesis testing at significance level a of 0.05, the test statistic iscalculated as 2. If P(X >2) = 0.03, then we have sufficient evidence to reject the nullhypothesis.Refer to the following frequency distribution for Questions 2, 3, 4, and 5. Show all work. Just theanswer, without supporting work, will receive no credit.A random sample of 100 students was chosen from UMUC STAT 230 classes. Thefrequency distribution below shows the distribution for study time each week (in hours).Study Time (in hours) Frequency Relative Frequency0.0 4.9 55.0 9.9 1310.0 14.90.2215.0 -19.9 42Total 10020.0 24.9Refer to the following information for Questions 6, 7, and 8. Show all work. Just the answer,without supporting work, will receive no credit.A fair 6-faced die is rolled two times. Let A be the event that the outcome of the first roll is amultiple of 3, and B be the event that the outcome of second roll is greater than 4.first roll is a multiple of 3? (10 pts)Refer to the following situation for Questions 9, 10, and 11.The five-number summary below shows the grade distribution of two STAT 230 quizzes.Minimum Q1 Median Q3 MaximumQuiz 1 12 40 60 95 100Quiz 2 20 35 50 90 100For each question, give your answer as one of the following: (a) Quiz 1; (b) Quiz 2; (c) Both quizzes have the same value requested; (d) It is impossible to tell using only the given information. Then explain your answer in each case.Refer to the following information for Questions 12 and 13. Show all work. Just the answer,without supporting work, will receive no credit.There are 1000 juniors in a college. Among the 1000 juniors, 300 students are takingSTAT230, and 150 students are taking PSYC300. There are 50 students taking bothcourses.courses? (10 pts)he/she is taking PSYC300? (10 pts)(a) Construct a table describing the probability distribution.(b) Determine the mean and standard deviation of x. (Round the answer to two decimal places)garden which will be ready to harvest in about 10 days. Based on her experience, the probabilityof a cucumber being eaten by the rabbits before harvest is 0.60. Show all work. Just the answer,without supporting work, will receive no credit.(a) Let X be the number of cucumbers that Mimi harvests (that is, the number of cucumbers not eatenby rabbits). As we know, the distribution of X is a binomial probability distribution. What is thenumber of trials (n), probability of successes (p) and probability of failures (q), respectively?(b) Find the probability that Mimi harvests at least 2 of the 10 cucumbers. (round the answer to 3 decimal places)(c) How many cucumbers can she expect to harvest?Refer to the following information for Questions 17, 18, and 19. Show all work. Just the answer,without supporting work, will receive no credit.The heights of pecan trees are normally distributed with a mean of 10 feet and a standard deviation of 2 feet.The percentile of the pecan tree height distribution.scores have a population standard deviation of 300. Construct a 95% confidence interval estimateof the mean SAT scores. Show all work. Just the answer, without supporting work, will receiveno credit.015.0:pH 5.0:pHIn a random sample of 100 subjects, the sample proportion is found to be47.0P (a) Determine the test statistic. Show all work; writing the correct test statistic, withoutsupporting work, will receive no credit.(b) Determine the p-value for this test. Show all work; writing the correct P-value,without supporting work, will receive no credit.H at the0.01 level? Explain.(c) Is there sufficient evidence to justify the rejection ofa hypothesis test. A random sample of 5 UMUC students was chosen. The students tooka 30-minute exercise every day for 6 months. The weight was recorded for eachindividual before and after the exercise regimen. Does the data below suggest that theregular exercise helps weight loss?Weight (pounds)Subject Before After1 159 1302 170 1603 185 1804 165 1655 200 190Assume we want to use a 0.01 significance level to test the claim.(a) Identify the null hypothesis and the alternative hypothesis.(b) Determine the test statistic. Show all work; writing the correct test statistic, withoutsupporting work, will receive no credit.(c) Determine the p-value. Show all work; writing the correct critical value, withoutsupporting work, will receive no credit.(d) Is there sufficient evidence to support the claim that regular exercise helps weightloss? Justify your conclusion.Her null hypothesis and alternative hypothesis are:(a) Determine the test statistic. Show all work; writing the correct test statistic, withoutsupporting work, will receive no credit.(b) Determine the p-value for this test. Show all work; writing the correct P-value,without supporting work, will receive no credit.(c) Is there sufficient evidence to justify the rejection of H at the significance level of00.05? Explain.