Question 1:
Manufacturing companies strive to maintain production consistency, but it's often difficult for outsiders to tell whether they've succeeded. Sometimes, however, we can find a simple sample. The candy company that makes M&M's claims that 10% of the candies it produces are green and that bags are packed randomly. We can check on its production controls by sampling bags of candies. Suppose we open bags containing about 50 M&M's and record the proportion of green candies.
a) If we plot a histogram showing the proportions of green candies in the various bags, what shape would you expect it to have?
b) Can that histogram be approximated by a Normal model? Explain
c) Where should the centre of the histogram be?
d) What should the standard deviation of the proportion be?
Question 2:
According to the Canadian Radio-television and Telecommunications Commission (CRTC), in 2008, 18% of Anglophone Canadians owned a video-enabled cell phone, but only 5% viewed video on their cell phones. You take a random sample of 285 anglophone Canadians in order to interview them about their use of cell phone video.
a) What is the probability that more than 20% of them owned a video-enabled cell phone in 2008?
b) What is the probability that less than 4% of them viewed video on their cell phone in 2008?
c) Suppose 23% of your sample owns a video-enabled cell phone today. Does that make you think the overall percentage of Anglophone Canadians owning a video0enabled cell hone has increased since 2008? State your reasons clearly.
d) Suppose 6% of your sample views video on their cell phones today. Does that make you think the overall percentage of Anglophone Canadians viewing video on their cell phones has increased since 2008? State your reasons clearly.
Question 3:
A biotechnology firm is planning its investment strategy for research into human cloning. A poll found that only 8% of a random sample of 1012 Canadian adults approved of attempts to clone a human.
a) Find the margin of error for this poll if we want 95% confidence in our estimate of the percentage of Canadian adults who approve of cloning humans.
b) Explain what that margin of error means.
c) If we only need to be 90% confident, will the margin of error be larger or smaller? Explain.
d) Find that margin of error.
Question 4:
Angus Reid Strategies surveyed 1016 Canadian adults (assume a 50/50 split between men and women) and found that 65% of men and 45% of women support scrapping the penny.
a) Construct a 90% confidence interval for the difference in support for scrapping the penny between men and women.
b) Construct a 99% confidence interval for the difference in support between men and women.
c) Interpret the meaning of the fact that one of these confidence intervals is wider than the other.
Question 5:
In 2008, a Gallup Poll asked 2336 U.S. adults aged 18 or over how they rated economic conditions. In a poll conducted from January 27 through February 1, 2008, 24% rated the economy as "Excellent/Good." A recent media outlet claimed that the percentage of Americans who felt the economy was in "Excellent/Good" shape was, in fact, 28%. Does the Gallup Poll support this claim?
a) Find a 95% confidence interval for the sample proportion of U.S. adults who rated the economy as "Excellent/Good." Check conditions.
b) Does your confidence interval provide evidence to support the claim?
c) What is the significance level of the test in part b)? Explain.
Question 6:
The same Gallup Poll data from Question 5 also reported that 33% of those surveyed rated the economy as "Poor." The same media outlet claimed the true proportion to be 30%. Does the Gallup Poll support this claim?
a) Find a 95% confidence interval for the sample proportion of U.S. adults who rated the economy as "Poor." Check conditions.
b) Does your confidence interval provide evidence to support the claim?
c) What is the significance level of the test in part b)? Explain.
Question 7:
A company with a large fleet of cars hopes to keep gasoline costs down and sets a goal of attaining a fleet average of at least 26 miles per gallon. To see if the goal is being met, the company checks the gasoline usage for 50 company trips chosen at random, finding a mean of 25.05 mpg and a standard deviation of 4.83 mpg. Is this strong evidence that the company has failed to attain its fuel economy goal?
a) Write appropriate hypotheses.
b) Are the necessary assumptions to perform inference satisfied?
c) Test the hypothesis and find the P-value
d) Explain what the P-value means in this context.
e) State an appropriate conclusion.
Question 8:
During 2008-11, Via Rail installed new passing tracks and a Centralized Traffic Control system on the line between Kingston and Ottawa in order to reduce delays. Suppose that, before the improvements, trains between Kingston and Ottawa were on average 12.4 minutes late. After the improvements, a consumer group monitored delays during a period of three months. For a random sample of 35 trains, it found the average delay to be 6.2 minutes with a standard deviation of 8 minutes. "The improvements reduced delays by half," they claimed.
a) Have average delays been reduced? State your hypotheses and calculate a P-value. State any assumptions you're making and whether they're likely to be true.
b) Calculate the 95% confidence interval for the delays after the improvements, assuming the conditions are satisfied. Comment on validity of your confidence interval.
c) Comment on the ethics of the consumer group's statement.