Question

1. Short answer only.

a. Suppose that we tell you that we flipped a coin multiple times and it landed heads 75% of the time. Would you be reasonably convinced that this was not a fair coin (where “fair” means that the coin has a .5 probability of landing “heads”)? If so, explain why. If not, describe what additional information you would ask for and explain why it is necessary.

b. A college professor asked students in his class to report how many miles they were from where they were born. The distribution was sharply skewed to the right, with some very large outliers. How would the mean distance compare to the median distance? Explain briefly.

c. Suppose you conducted a hypothesis test and decided to reject the null hypothesis at the α=.01 significance level. If you decided instead to use the α=.02 significance level, would you reject the null hypothesis, fail to reject the null hypothesis, or do you not have enough information to say? Explain briefly

d. Suppose the IQs of San Luis Obispo residents followed a normal distribution, with mean 105 and standard deviation 12. Which is more likely: that a randomly selected resident will have an IQ greater than 120 or that the average IQ in a random sample of 10 residents will exceed 120? Explain briefly. (You do not need to perform any calculations.)

e. In an experiment that compared acupuncture to conventional therapy for treating patients with chronic low-back pain, a 95% confidence interval for the difference in population proportions experiencing improvement was calculated to be (.135, .265). Does this suggest the difference in improvement proportions between the two groups is statistically significant? Explain briefly.

2. (8 points) Statisticians analyzed a huge set of data about people who were sent a mass mailing solicitation (request for donations) from a nonprofit organization. The observational units were the recipients of the mailing. 

The two response variables of interest were: Whether the person sent a donation How much the person donated Hundreds of explanatory variables were recorded about these mailing recipients. These included: Age Whether the person owned a home, rented a home, or did not answer that question How many mailings the person received in the previous 12 months Whether the person donated in the previous 12 months How much the person donated in the previous 12 months 

a. State one research question (make up your own, based on these variables) that could be investigated with these data, for which comparing boxplots would be a reasonable technique. Your research question can lead to a comparison of two boxplots or more than two boxplots.

b. State one research question (make up your own, based on these variables) that could be investigated with these data, for which a scatterplot would be a reasonable graphical display to explore your question. 

3. (4 points) Suppose that 85% of the email messages arriving to a college’s computer system are spam. You want to take a random sample of incoming email messages and perform probability calculations involving the sample proportion that are spam. How large must the sample size be so you can reasonably apply the Central Limit Theorem?

5. (20 points) It has been conjectured that two-thirds of all students have an active, as opposed to reflective, learning style. Data are available from a large sample of introductory statistics students at a public university in the eastern United States, who took a survey that assessed their learning style as active or reflective. Of the 962 students who participated, 596 were diagnosed as active learners and 366 as reflective learners. Use these sample data to conduct a significance test of the conjecture that two-thirds of all students have an active learning style. 

Report the hypotheses, test statistic, and pvalue. Include a check of technical conditions. Also indicate your test decision at the α=0.05 significance level and summarize your conclusion in context.

Answer 1

1.a No , I wouldn't be convinced that it is a fair coin , because to conclude that it is a fair experiment of tossing of coin should be repeated sufficiently large number of times with probabilty of success of obtaining head is 0.5 by Law of Large Numbers to conclude that it is a fair coin.

1.b As the distribution of the distance where student born is skwed to the right side, we can conclude that it is positively skewed .Which means that , the frequency of students with large distance from the place where they born is high as compared to the students with small distance.Thus mean of the data is more than the median ,in other words mean is located on the right hand side of median .