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STAT 200 - Practice Exam 2
Solutions appear at the end of the document. (Don’t forget to also look over Homework assignments.)
Question 1 For each of the following scenarios, identify the following: i. the random variable of interest, ii. whether the random variable is discrete or continuous, iii. the name of the distribution that describes the behavior of the random variable iv. the true or estimated values for the parameters associated with that distribution v. the expected value and variance of the random variable
a) A police department observes that 20% of all drivers come to a complete stop at an intersection having flashing red lights in all directions when no other cars are visible. The police department will random choose 20 drivers in the next hour and observe whether they come to a complete stop.
b) A certain T.A. never finishes his lecture before the bell rings to end the period and always finishes his lecture within 2 minutes after the bell rings.
Question 2 The heights of a large sample of men are close to normally distributed with an average of 70 inches and a SD of 2. inches.
a) About 68% of the heights fall between ______ inches and ______ inches.
b) About 95% of the heights fall between ______ inches and ______ inches.
c) Convert 73 inches to standard units. (How many SD above average is 73 inches?)
d) What percentage of men are between 70 and 73 inches tall?
e) How tall is a man who is at the 95th^ percentile?
The scores on a final exam closely follow the normal curve with an average of 70 and a SD of 10.
a) Convert a score of 83 to standard units.
b) What percentage of the students scored over 83?
c) The professor wants to give the top 20% of the students A’s, the next 30% B’s, and the rest C’s and lower. What final score should be a cutoff between an A and B?
d) Which final score should be a cutoff between a B and C? (No need to show work) ______
Question 4 Suppose a fair coin is tossed 400 times.
a) What is the expected number of heads? ______
b) What is the SE for the number of heads? (Recall that the SD for one coin flip is 0.5.)
c) Use the normal approximation to find the chance of getting over 220 heads.
d) The chance of getting exactly 200 heads is closest to:
Circle one: 1% 4% 8% 50%
In a certain city, 20% of the adults own a cell phone. A simple random sample of 900 adults is drawn.
a) What are the expected value and standard error for the percent of the sample that own a cell phone?
Expected value:
Standard error:
b) Use the normal approximation to estimate the chance that between 18% and 22% of the sample owns a cell phone.
Question 9 As part of a survey on sleeping habits of UI undergraduates, a simple random sample of 100 UI undergraduates are polled. The 100 students report an average of 7 hours of sleep per night with a SD of 1 hour. The data are not normally distributed.
a) We can estimate the average number of sleeping hours among all UI undergraduates to be 7 hours, give or take _____ hours or so.
Circle one: i) 0.1 ii) 0.01 iii) 1
b) Indicate whether each of the statements below is true or false:
True False i) 7 ± 0.2 is a 95% confidence interval for the average number of hours
all UI undergraduates sleep per night □^ □
ii) About 95% of the students in the survey reported sleeping between 7
± 0.2 hours per night □^ □
iii) About 95% of the students in the survey reported sleeping between 7
± 2.0 hours per night □^ □
iv) About 95% of UI undergraduates get between 7 ± 0.2 hours of sleep
per night □^ □
v) 7 ± 0.2 is a 95% confidence interval for the average number of hours
all American students sleep per night □^ □
Mark likes to amaze his friends with his imbalanced coin. On a given flip, the chance of heads is 2/3 and the chance of tails is 1/3.
Throughout the summer, Mark will ask 100 people to each flip his imbalanced coin 100 times to guess the probability of heads. Each person will construct a 95% confidence interval for the true probability of heads based on his (or her) experimental results.
a) How many of the 100 confidence intervals would you expect to miss the true probability of the imbalanced coin? (That is, how many of the confidence intervals would you expect to not contain the value 2/3?)
b) Suppose Molly’s 100 flips resulted in 60 heads and 40 tails. Construct her 95% confidence interval for the true probability (i.e. percentage) of heads.
Question 11 Forbes Magazine surveyed 1,511 adults in the United States to learn about their spending plans for the upcoming holiday season. The results are summarized below:
- 36% plan to spend less money than last year
- 46% plan to spend about the same amount of money as last year
- 18% plan to spend more money than last year
(You may assume that the 1,511 adults constituted a simple random sample of all U.S. adults.)
a) Based only on the results of this poll, what is your best guess for the percentage of all U.S. adults who plan to spend more this holiday season than last year?
b) Construct a 95% confidence interval for the percentage of all U.S. adults who plan to spend more this holiday season than last year.
Each container of Aquafina bottled water claims to contain 500mL (16.9 fluid ounces) of liquid. A consumer watchdog group suspected that Aquafina’s packaging process is flawed and is not abiding by laws which state Aquafina must provide a “full” bottle of water, and not one which is too far under or over the advertised amount. So, they conducted a multistage cluster sample and collected 100 bottles of Aquafina from supermarket shelves across the country. These bottles averaged only 490mL of water with a standard deviation of 40mL. (You may assume the distribution of individual volume is approximately normal.)
We wish to test whether the observed shortage can be explained by chance, or if Aquafina’s packaging process is truly flawed.
a) State the null and alternative hypotheses in your own words.
b) Assuming the null hypothesis is true, then the average volume of a bottle of Aquafina is ________mL.
c) Calculate the critical value(s) under the null hypothesis. (i.e. Find the cutoff values which define the limits of your rejection region.)
d) Calculate the test statistic for testing whether or not the null hypothesis is true.
d) Find the p -value for testing whether or not the null hypothesis is true.
e) What is the most reasonable decision for this test?
Choose one: (i) Reject the null hypothesis
(ii) Fail to reject the null hypothesis
(iii) Accept the null hypothesis
f) State what conclusion can be made regarding the consumer group’s claim based on the results of this significance test.
g) Regardless of (e), compute the power for this sample which yielded an alternative of 490mL per bottle.
Are men or women more likely to believe in “love at first sight?” We asked a random sample of 500 men and 500 women if they had ever "fallen in love at first sight." 44% of the men and 36% of the women said yes.
a) State the null and alternative hypotheses in your own words.
b) Assuming the null hypothesis is true, then the difference in percentage between men and women is ______.
c) Calculate the estimated population proportion of those people who believe in love at first sight, and estimate the standard error.
d) Calculate the test statistic for testing whether or not the null hypothesis is true.
e) Find the p -value for testing whether or not the null hypothesis is true.
f) What is the most reasonable conclusion from this test?
Choose one: (i) Fail to reject the null hypothesis
(ii) Reject the null hypothesis
g) State the conclusion as it pertains to the actual experiment.
a) (i) 0. b) i) True ii) False iii) False iv) False v) False
Question 10 a) 5 b) (50.2%, 69.8%) c) (ii) The SE will be 2 times larger.
Question 11 a) 18% b) (16%, 20%)
Question 12 a) H 0 : Weight difference is due to random chance, or μ = 16 HA: Weight difference is due to the weight-loss pill, or μ < 16 b) 16 ounces c) -1. d) 0. e) (i) Fail to reject the null hypothesis f) There is not sufficient evidence to conclude that the weight-loss pill is effective, or The weight difference is most likely just due to random chance, or The data suggests that the weight-loss pill is not effective. g) Critical value at α = 0.05 level is 15. Power = 0.
Question 13 a) H 0 : Volume difference is due to random chance, or μ = 500 HA: Volume difference is due to flawed packaging, or μ ≠ 500 b) 500mL c) At α = 0.05 level, reject if z < -2 or z > 2, that is if X < 492 or X > 508. d) -2. e) (i) Reject the null hypothesis f) There is sufficient evidence to conclude that the packaging process is flawed, or The volume difference is most likely just due to flawed packaging, or The data suggests that the packaging process is flawed. g) Power = 0.
Question 14 a) H 0 : Difference is due to random chance, or μM - μW = 0 HA: Difference in proportion is due to difference between men and women, or μM - μW ≠ 0 b) 0% c) p = 0.40, SE = 0. d) z = 2. e) p -value=0. f) (ii) Reject the null hypothesis g) There is highly significant evidence to conclude that there is a difference between men and women with to falling in love at first sight.
