3/24/2016 Stats 250 W16 HW 7 Closing date: 3/24/16 5:00PM 56ee06d0a938da0b00f665 Question 1 Background : Space Lab Life Science Experiment In an experiment 28 male rats were randomly assigned to one of two groups. The 14 male rats in Group 1 were sent into space. Upon their return, their red blood cell mass (in milliliters, mL) was determined. The 14 male rats in Group 2 (the control group) were held in the science lab at Cape Kennedy. Unfortunately, one of the rats in the control group died during the experiment, so the sample sizes were 14 and 13, respectively. The descriptive statistics were obtained at the conclusion of the mission and are provided in the table. The data analyst would like to use the data to produce a 95% condence interval for the dierence in the population mean red blood mass, 1 - 2. Question 1 Subquestions 1.a 0.5 point(s) After examining the two sample standard deviations, the analyst decided to also look at Levene's test result to help assess if the population variances can be considered equal. The p-value for Levene's test was 0.055. Using the 10% signicance level for Levene's test, which technique is reasonable to use for making the condence interval estimate? Pooled 95% condence interval General (unpooled) 95% condence interval 1.b 0 point(s) Produce the corresponding 95% condence interval for the dierence in the population mean red blood cell mass, 1 - 2. Show your work. Interpret your interval in context. Student answer No answer entered. 1.c 2 point(s) 1/11 3/24/2016 The analyst observes that the value of 0 does fall in the 95% condence interval. As a result of this 95% condence interval, which conclusion is appropriate at the 5% level of signicance? It appears the population mean red blood cell mass for male rats sent into space is less than the population mean red blood cell mass for male rats not sent into space. There does not appear to be a dierence between the population mean red blood cell mass for male rats sent into space and the population mean red blood cell mass for male rats not sent into space. It appears the population mean red blood cell mass for male rats sent into space is more than the population mean red blood cell mass for male rats not sent into space. 1.d 0 point(s) Which of the following is/are additional conditions required for the condence interval to be valid? Select all required conditions. The two samples of red blood cell mass measurements for the two sets of male rats can be considered random samples. The two samples of red blood cell mass measurements for the two sets of male rats can be considered independent samples. The red blood cell mass measurements for each of the two populations of male rats (Space Flight versus Control) follow a normal distribution. The dierence in red blood cell mass measurements (dierence = Space Flight minus Control) for the population of all male rats follows a normal distribution. The total sample size (the total number of male rats in the experiment) must be larger (at least 25). None of the above. Question 2 : 1 point(s) Common Population Standard Deviation A psychologist is conducting a study to learn about the dierence between two population means, 1 - 2 using a two-independent samples t procedure. The two sample standard deviations are s1 = 4.0 points and s2 = 3.0 points and the sample size for the group 1 was more than the sample size for group 2. Which value is the only reasonable value for the pooled estimate of the common population standard deviation? Question 2 Multiple Choice Options 2/11 3/24/2016 2.3 points 3.3 points 3.5 points 3.7 points 4.9 points Cannot be determined. Question 3 Background : Cash Offer for a Used Car A consumer organization decided to assess if the age of a used car owner has any eect on the size of a cash oer for a used car. In particular, they hypothesize that dealerships oer more to middle-aged car owners as compared to young car owners, on average. A 5% signicance level was selected. The study involved 12 young people (6 men and 6 women all between 18 and 24 years old, referred to as group 1) and 12 middleaged people (6 men and 6 women all between 36 and 45 years old, referred to as group 2). These 24 people acted as the \"owner\" of a specic 6-year-old used car. A total of 24 dierent car dealerships were selected at random from those in the Detroit area. Each subject in the study went to one car dealership to obtain a cash oer for the same used car. The resulting cash oers in hundreds of dollars are available in the data set called CashOer.RData (https://pbj-coursework.s3.amazonaws.com/Intro%20to%20Statistics/instructor/11-192015/2_53_27-18_42_774-CashOer.RData). Question 3 Subquestions 3.a The null hypothesis can be stated as the alternative hypothesis 0.5 point(s) . Which of the following is the appropriate direction for _____ ? greater than (>) less than (<) not equal to 3.b 0 point(s) Consider the following incorrect denition for the rst population mean: 3/11 3/24/2016 1 = the sample mean cash oer for the young used car owners How should you complete this statement to improve this denition? Change the word _______________ to _______________. Student answer No answer entered. 3.c 2 point(s) Use R to perform the two-sample t-test. Generate both the pooled and unpooled (general) two independent Samples test results. Copy and paste both sets of test results below OR save the test outputs as a jpg and upload it. Student answer No answer entered. 3.d 0 point(s) To help you decide which of the two test results to use, you suggest that Levene's test be conducted. Which of the following is the appropriate null hypothesis for Levene's test stated in words? The two sample variances are equal. The two sample variances are similar. The two population variances are equal. The two population variances are similar. 3.e 0 point(s) Here are the results for Levene's test. Rcmdr> leveneTest(cashoer ~ ownerage, data=CashOer, center="mean") Levene's Test for Homogeneity of Variance (center = "mean") Df F value Pr(>F) group 1 0.0462 0.8318 22 Based on these results, which t-test results will you report? Two independent samples general (unpooled) t-test Two independent samples pooled t-test 4/11 3/24/2016 3.f 0 point(s) Report the t-test statistic value, and the p-value for the hypothesis in part (a). Student answer No answer entered. 3.g 0 point(s) What is the distribution that would be used to nd the p-value for the t-test? N(0,1) t(22) t(23) t(21.997) t(11) 3.h 0 point(s) Clearly state the appropriate conclusion to this investigation in context. Student answer No answer entered. 3.i 0 point(s) If there really was no eect of the age of the used car owner on the mean cash oer (for the two populations), what would be the expected value of the t-test statistic? 0 0.05 1.96 -3.20 5/11 3/24/2016 cannot tell as we do not know the values of the two population means 3.j 0 point(s) A summer intern who was helping on this project for the consumer organization remembered that there was a normal model condition required for conducting this test. He created the following graph for assessing that normality condition. Clearly explain in one sentence why this is not the appropriate graph to examine and suggest what should be examined instead. Make sure to include context. Student answer No answer entered. 6/11 3/24/2016 Question 4 Background : Atlantic vs Pacic Seagulls An ecologist wishes to compare the feeding habits of seagulls at the Atlantic coast with seagulls at the Pacic coast as to daily food consumption (in grams). He would like to assess if Atlantic seagulls (group 1) eat more on average than the Pacic seagulls (group 2). Thus, the hypotheses to be tested at a 5% level of signicance are H0: 1 = 2 versus Ha: 1 > 2 . He obtains the data from randomly selected seagulls on each of the two coasts. These seagulls were captured at dusk and fed the following day to measure food consumption while caged during the following 24 hours. Numerical summary measures are provided after looking at appropriate graphical summaries rst. Question 4 Subquestions 4.a 0.5 point(s) Given that the two sample standard deviations seems quite similar, the ecologist decides to use the pooled twosample t-test. Using the two sample standard deviations, the estimate of the common population standard deviation was 3.804 grams. What is the value of the test statistic? 1.20 1.367 1.24 3.804 4.b 0 point(s) The resulting p-value is 0.11. Which of the following is the appropriate decision and conclusion? Reject the null hypothesis and conclude the average food intake for the population of all Atlantic seagulls is greater than that for the population of all Pacic seagulls. Reject the null hypothesis and conclude there was insucient evidence to say the average food intake for the population of all Atlantic seagulls is greater than that for the population of all Pacic seagulls. Fail to reject the null hypothesis and conclude that the average food intake for the population of all Atlantic seagulls is greater than that for the population of all Pacic seagulls. 7/11 3/24/2016 Fail to reject the null hypothesis and conclude there was insucient evidence to say the average food intake for the population of all Atlantic seagulls is greater than that for the population of all Pacic seagulls. Question 5 Background : Changing Temperatures A meteorologist wants to determine if it was colder this past October 2013 than it was in October of the previous year 2012, on average, using a 10% signicance level. From each month, he randomly selects 8 dierent days from each year and records the temperature. Let Group 1 = temperatures from October this year 2013 and Group 2 = temperatures from October last year 2012. Question 5 Subquestions 5.a 0.5 point(s) Which of the following is the appropriate symbol to complete set of hypotheses? H0: 1 = 2 versus Ha: 1 ? 2 > (greater than or one-sided to the right) < (less than or one-sided to the left) not equal to (two-sided) 5.b 0 point(s) The meteorologist calculates the sample standard deviations for the two sets of temperature measurements to be s1 = 3.5 degrees and s2 = 11.6 degrees, respectively. Based on these sample standard deviations, should the meteorologist use a pooled or unpooled t-test? Pooled t-test Unpooled (or General) t-test 5.c 2 point(s) To verify the necessary data conditions before conducting the hypothesis test, the meteorologist's software program makes two QQ plots - one for each set of temperatures from the two October months. What if the points on the plots don't follow the lines in the QQ plots, that is, if the distributions of temperatures for each population appear to be strongly skewed? Should the meteorologist continue with the two-sample t-test? Yes 8/11 3/24/2016 No 5.d 0 point(s) Suppose the meteorologist conducts the appropriate test and obtains a p-value of 0.076 and thus decides to reject the null hypothesis. What type of error could he have made? Type 1 error Type 2 error Since the decision has been made, no error is possible. Question 6 Background : Vitamin C and the Common Cold An experiment was planned to assess if Vitamin C really does help cure the common cold more quickly. A random sample of 70 adults were randomized to one of two treatment groups. At the onset of a cold, subjects assigned to Group 1 received a placebo dose of vitamin C while subjects assigned to Group 2 received a daily dose of 4 mg of vitamin C. The treatment started on the day when the cold was rst acknowledged by each person (onset) and continued until the cold symptoms were resolved. The response recorded for each subject was recovery time, measured in the number of days between onset and resolution. The researchers set the signicance level to 5%. Question 6 Subquestions 6.a 0.5 point(s) Let 1 = the population mean recover time (in days) for all adults with a cold taking the placebo; whereas 2 has a similar denition for the Vitamin C treated population. Given the researcher was interested in assessing that Vitamin C would reduce the population mean recovery time, what symbol should be used to complete the alternative hypothesis? H0: 1= 2 versus Ha: 1 _____2 > (greater than) < (less than) != (not equal to) 6.b 0 point(s) 9/11 3/24/2016 Although 35 people were randomized to each treatment group at the start, several subjects elected to withdraw from the study. The resulting summary statistics for the 62 subjects that completed the study are provided. The researchers rst took note that the sample standard deviations of 2.9 days and 1.2 days are quite dierent. Thus they elected to use the general (unpooled) approach. Provide the estimate for the dierence in the population means (that is report the dierence in the sample means, 1 - 2), and compute the corresponding standard error of that estimate. Show your work and include your units). Student answer No answer entered. 6.c 2 point(s) Compute the (unpooled) t test statistic value. Student answer No answer entered. 6.d 0 point(s) If we were conducting this general (unpooled) test without a statistical package, the conservative (and approximate) degrees of freedom can be found as the minimum of (n1 -1) and (n2 -1), namely the minimum of 29 and 31, or 29 df. However, some computer packages would use a Welch's formula to compute a less conservative degrees of freedom and would set the df = 38.14 (which is larger than 29). With this in mind, determine if the following statement is true or false. "In general, a larger degrees of freedom would lead to a more powerful hypothesis test." True False 6.e 0 point(s) The statistical computer package would give a corresponding p-value of 0.031. Clearly state the decision (at a 5% signicance level) and write an appropriate conclusion (in the context of the problem). Student answer No answer entered. 10/11 3/24/2016 Edit Assignment (/#!/students/56d5a41ece99240b0000fc26/do) 56ee06d0a938da0b00f665 11/11