1. It is 1581 Anno Domini. At the Undergraduate School of UMUC, besides Assistant Academic Director of Mathematics and Statistics, I am also the Undergraduate School-appointed CPA, Coffee Pot Attendant. It is a very important office sponsored by the Holy See. I have taken this job very seriously, because I believe that I am the key to increased productivity at the Undergraduate School. Why, by mid-morning, many of my colleagues act as if they were. It is imperative that I restore productivity via a secret naturally-occurring molecule, caffeine…….. In order to see if my secret molecule works, I have observed the time, in hours, a random selection of ten of my colleagues who could stay awake at the extremely long-winded Dean’s meeting as soon as it started. Oh, yes, one fell asleep even before the meeting started!
1.9 0.8 1.1 0.1 -0.1
4.4 5.5 1.6 4.6 3.4
Now, I have to complete a report to the Provost’s Office on the effectiveness of my secret molecule so that UMUC can file for a patent at the United Provinces Patent and Trademark Office as soon as possible. But I need the following information:
What is a 95% confidence interval for the time my colleagues can stay awake on average for all of my colleagues? Was my secret molecule effective in increasing their attention span, I mean, staying awake? And, please explain…..
OK, take your time, but hurry up!!!
4. Why is a 99% conﬁdence interval wider than a 95% conﬁdence interval?
12. A person claims to be able to predict the outcome of ﬂipping a coin. This person is correct 16/25 times. Compute the 95% conﬁdence interval on the proportion of times this person can predict coin ﬂips correctly. What conclusion can you draw about this test of his ability to predict the future?
15. You take a sample of 22 from a population of test scores, and the mean of your sample is 60. (a) You know the standard deviation of the population is 10. What is the 99% conﬁdence interval on the population mean. (b) Now assume that you do not know the population standard deviation, but the standard deviation in your sample is 10. What is the 99% conﬁdence interval on the mean now?
18. You were interested in how long the average psychology major at your college studies per night, so you asked 10 psychology majors to tell you the amount they study. They told you the following times: 2, 1.5, 3, 2, 3.5, 1, 0.5, 3, 2, 4. (a) Find the 95% conﬁdence interval on the population mean. (b) Find the 90% conﬁdence interval on the population mean.
100. What is meant by the term “90% confident” when constructing a confidence interval for a mean? a. If we took repeated samples, approximately 90% of the samples would produce the same confidence interval.
b. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the sample mean.
c. If we took repeated samples, approximately 90% of the confidence intervals calculated from those samples would contain the true value of the population mean.
d. If we took repeated samples, the sample mean would equal the population mean in approximately 90% of the samples.
106. Suppose that a committee is studying whether or not there is waste of time in our judicial system. It is interested in the mean amount of time individuals waste at the courthouse waiting to be called for jury duty. The committee randomly surveyed 81 people who recently served as jurors. The sample mean wait time was eight hours with a sample standard deviation of four hours.
a. i. x ¯ = __________ ii. sx = __________ iii. n = __________ iv. n – 1 = __________
b. Define the random variables X and X ¯ in words.
c. Which distribution should you use for this problem? Explain your choice.
d. Construct a 95% confidence interval for the population mean time wasted. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound.
e. Explain in a complete sentence what the confidence interval means.
112. In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Assume the underlying distribution is approximately normal.
a. Which distribution should you use for this problem? Explain your choice.
b. Define the random variable X ¯ in words.
c. Construct a 95% confidence interval for the population mean cost of a used car. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound.
d. Explain what a “95% confidence interval” means for this study
116. What is the error bound?
120. An article regarding interracial dating and marriage recently appeared in the Washington Post. Of the 1,709 randomly selected adults, 315 identified themselves as Latinos, 323 identified themselves as blacks, 254 identified themselves as Asians, and 779 identified themselves as whites. In this survey, 86% of blacks said that they would welcome a white person into their families. Among Asians, 77% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.
a. We are interested in finding the 95% confidence interval for the percent of all black adults who would welcome a white person into their families. Define the random variables X and P′, in words.
b. Which distribution should you use for this problem? Explain your choice.
c. Construct a 95% confidence interval. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound
130. On May 23, 2013, Gallup reported that of the 1,005 people surveyed, 76% of U.S. workers believe that they will continue working past retirement age. The confidence level for this study was reported at 95% with a ±3% margin of error.
a. Determine the estimated proportion from the sample.
b. Determine the sample size.
c. Identify CL and α.
d. Calculate the error bound based on the information provided.
e. Compare the error bound in part d to the margin of error reported by Gallup. Explain any differences between the values.
f. Create a confidence interval for the results of this study.
g. A reporter is covering the release of this study for a local news station. How should she explain the confidence interval to her audience?