HP340: Statistical Methods, Spring 2015Due Date: 3:30PM, Tuesday Feb 17, 2015 Homework 4Total Points: 50Mark the letter of the correct answer for multiple-choice questions. Show your work/explain yourreasoning when calculations are required. Late assignments will not be accepted.1. (6 points) You roll a pair of fair 6-sided dice.a) (1 point) What is the total number of possible outcomes of this probability experiment?36 outcomes Assuming all outcomes are equally likely:b) (1 point) What is the probability of getting a 5 AND a 6? 1/6(1/6)+1/6(1/6)=1/18c) (2 points) What is the probability of getting one even AND one odd number? ½(1/2)=1/4d) (2 points) What is the probability of getting either two odd numbers OR two even numbers? ½(1/2)+1/2(1/2)= 1/22. (19 points total) a) (2 points) To obtain an accurate estimate of a parameter, a sample should be ____.A. small but with high variability B. consistent but with high variabilityC. a non-random sample of the populationD. large but not representative of the population E. representative of the population.b) (2 points) The _____a score vary in the population and the _____ the sample size, then the more accurately a sample mean will estimate a population mean.A. less ; smallerB. more ; largerC. less ; largerD. more ; smallerE. Cannot be determined without knowing the exact sample size.c) (5 points) True or False. Consider two samples from the same population. Sample 1 has size N=200 and sample 2 has size N=500.A. The sampling distribution of the mean based on sample 1 will have a larger standard deviation than the sampling distribution of the mean based on sample 2. TrueB. The standard error of the mean based on sample 1 will be larger than the standard error of the mean based on sample 2. TrueC. A 95% confidence interval (CI) based on sample 1 is expected to be shorter than a 95% CI based on sample 2. FalseD. The mean of sample 2 is expected to be closer to the population mean than the meanof sample 1. TrueE. Sample 1 is expected to have a smaller mean than sample 2. Falsed) (7 points) True or False. Consider a p% confidence interval (CI) for the population mean constructed from a sample of size N. (e.g. if p=95 it's a 95% CI and if p=99 it’s a 99% CI).A. It will always contain the population mean. TrueB. It will always contain the sample mean. True C. There is a p% probability that the p% CI contains the population mean. FalseD. There is a p% confidence that the p% CI will contain the population mean. TrueE. There is a p% probability that the p% CI will contain the sample mean. TrueF. If one constructed a very large number of p% confidence intervals based on different samples of size N from the same population, about p% of the time the CIs will contain the population mean. TrueG. If one constructed a very large number of p% confidence intervals based on different samples of size N from the same population, about (100-p)% of the time the CIs will NOT contain the population mean. Falsee) (1 point) The estimated standard error of the mean is equal to _____.A. s2/NB. s/NC. s/ND. s2/NE. /N f) (2 points) If s = 15 and sX = 5 then N must be equal to _______. a. 3b. 9c. 5d. 154. (7 points) A population has a mean of 90 and a standard deviation of 10. If many samples of size N=100 are randomly drawn from this population, approximately what proportion of the sample means will fall between 89 and 91? Explain your answer and show your workz scores:(89-90)/10=0.1(91-90)/10=0.1P(89<p<91)=P(0.1<p<0.1)The table in the book shows the probability associated with a random variable of 0.1 is 0.4602 and 0.5398  0.5398-0.4602=0.0796 or 7.96% of the sample means will fall between 89 and 91.5. (18 points total) It has been extensively demonstrated that adequate night sleep is crucial for cognitiveperformance. To investigate whether first year USC students are getting adequate sleep the night before a final exam, 87 randomly selected first year USC students were asked to record their number of hours ofsleep the night before their final exams. The mean number of hours of sleep among the 87 students was6.2 with standard deviation of 1.5 hours. a. (3 points) Identify the population, the sample, and the parameter of interest in this study.Population: USC StudentsSample: randomly selected first-year USC studentsParameters: 87b. (5 points) Construct a 95% confidence interval for the mean number of hours first year USC students sleep the night before an exam. Explain the meaning of this confidence interval (you are expected to explain what this confidence interval says about the parameter of interest)X-1.96(meanx) to X+1.96(meanx) (meanx)=mean/rootN= 0.16085.8848 to 6.5152If I were to continuously sample from the 87 random USC students, and calculate the mean of hours of sleep they get the night before an exam and added/subtracted 1.96, there’s a 95% confidence that the calculated mean will fall between 5.88 and 6.51.c. (5 points) Construct now a 99% confidence interval for the mean number of hours first year USC students sleep the night before an exam. Explain the meaning of this confidence interval and contrast with the 95% confidence interval in part b.X-2.58 (meanx) to X+2.58(meanx)  6.02+/-0.41485.7851 to 6.6148If samples from the 87 USC students were taken and their mean hours of sleep the night before an exam was calculated and given +/-2.58, then theres 99% confidence that the calculated mean will lie between 5.78 and 6.61.The 99% confidence interval is much larger and allows more room then the 95% confidence which makes sense because the 99% confidence would be more accurate, and thus include more numbers.c. (5 points) Suppose that the 87 students represent a subset of a larger pool of 100 students randomly selected for the study. The 87 students are the ones who actually provided the investigators with their number of ours of sleep while the remaining 13 students did not respond when contacted by the investigators. If the 13 ‘non-responders’ were more likely to sleep fewer hours before an exam than a typical first year USC student, how would this affect the estimate and confidence interval for the mean number of hours first year USC students sleep the night before an exam?If there was a sample of students who most-likely slept less that were not included in the previous parameter, then the mean would decrease and the confidence intervals would shift to the left (towards smaller numbers) since