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STAT 3005: Statistical Methods
EXAM 2 (100 points + 10(bonus))
Monday, April 15, 2013
1 hour 1:25p.m. โ 2:25p.m
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This exam is broken into two sections: multiple-choice and open-ended. You do not
have to show your work for the multiple-choice section of the exam but you need to show
ALL your work for the open-ended questions.
SECTION 1: Multiple-Choice (5 points for each, except #6 6points)
( Note: Put your answer above the line )
- Which of the following is not correct? The standard deviation of the statistic describes ____ A. The standard deviation of the sampling distribution of that statistic. B. The standard deviation of the sample data measurements. C. How close that statistic falls to the parameter that it estimates. D. The variability in the values of the statistic for repeated random samples of size n.
- The central limit theorem (CLT) implies _____ A. All variables have approximately bell-shaped data distribution if a random sample contains at least about 30 observations. B. Population distributions are normal whenever the population size is large. C. For sufficiently large random samples, the sampling distribution of ๐ is approximately normal, regardless of the shape of the population distribution.
- The sampling distribution of a sample mean for a random sample size of 100 describes _____ A. How sample means tend to vary from random sample to random sample of size 100. B. How observations tend to vary from person to person in a random sample of size 100. C. How the data distribution looks like the population distribution when the sample size is larger than 30. D. How the standard deviation varies among samples of size 100.
- Increasing the confidence level causes the margin of error of a confidence interval to _____ A. Increases B. Decreases C. Stay the same
- The reason we use a z-score from a normal distribution in constructing a large-sample confidence interval for a proportion is that _____
D. decrease the sample size to 25.
SECTION 2: Open-ended
- (20 points) Your local sports team (Team A) is to play a set of seven games against another team (Team B). The outcome of a game is not dependent on the outcome of any other game and in any one game Team A has a 65% chance of beating Team B. Let X count the number of games that Team A wins.
a. What distribution does X follow? (Be sure to include the name of, and relevant parameters in, the distribution.) ๐ฟ~๐๐๐( ๐,. ๐๐)
b. Find the mean and the variance of X.
๐ฌ ๐ฟ = ๐๐ = ๐. ๐๐ = ๐. ๐๐ ๐ฝ๐๐ ๐ฟ = ๐๐๐ = ๐. ๐๐. ๐๐ = ๐. ๐๐๐๐ c. What is the probability that Team A wins five of the seven games? ๐ท ๐ฟ = ๐ = ๐๐ (. ๐๐)๐(. ๐๐)๐^ =.
d. Suppose that rather than playing all seven of the games versus Team B, the two teams only play as many games as it takes for either Team A or Team B to win exactly four games. Let X count the number of games it takes for four wins to occur for one the teams. Is X binomial? Justify.
No there are no longer a fixed number of trials, which is a requirement for a random variable to follow the binomial distribution.
- (20 points) When the 2000 GSS asked subjects if they would be willing to accept cuts in their standard of living to protect the environment, 344 of 1170 subjects said yes.
a. Estimate the population proportion who would answer yes. ๐ = (^) ๐๐๐๐๐๐๐ =. ๐๐๐
b. Find the margin of error for a 95% confidence interval for this estimate.
๐๐๐ = ๐๐โ๐๐^ ๐ฉ๐ช๐ง
= ๐. ๐๐.^ ๐๐๐๐๐๐๐^.^ ๐๐๐ =. ๐๐๐๐
c. Find a 95% confidence interval for that proportion. What do the numbers in this interval represent? ๐ ยฑ ๐๐โ๐ถ๐^ ๐๐๐
. ๐๐๐ยฑ. ๐๐๐๐ = (๐. ๐๐๐๐, ๐. ๐๐๐๐) We are 95% confident that the population proportion of people who are willing to cut their standard of living to protect the environment is between 26.79% and 32.01%.
d. State and check the assumptions needed for the interval in (c) to be valid.
**1. The observations are obtained randomly (through a simple random sample)
- The number of success and the number of failures are greater than or equal to 15.** ๐๐ = ๐๐๐ ๐๐ = ๐๐๐
- (6 points) A study at the Harvard School of Public Health found that 44% of 10, sampled college students were binge drinkers. A student at the University of Minnesota plans to estimate the proportion of college students at the school who are binge drinkers. How large a random sample would she need to estimate it to within 0.05 with 95% confidence, if before conducting the study she uses the Harvard study results as a guideline?
๐. ๐๐ ยฑ ๐. ๐๐ ๐.^ ๐๐
The average time of browsing the Internet of all the students in VT is
between 1.3921 and 2.7079 hours with confidence level 99%.
- (5 points) By setting the equation for the margin of error in a 100 1โ๐ผ)% confidence interval for a proportion to be at most some fixed value, say m, derive the formula for the minimal sample size, n, that is needed to guarantee a margin of error of at most m.
Good luck! ๏
