Question? 1
There are a number of different subway lines in London, some of which run in parallel under the same streets. The builders of the Underground arranged this by stacking the subway tunnels for each line under each other, sometimes two or three deep. At many Underground stations, to get to the deepest tunnels you take a long escalator ride. For instance, at the Pimlico station the escalator down to the deepest tunnel is like a moving stairway with 96 steps. During rush hour every single escalator step typically has two people on it side by side, so that the escalator had to be designed to carry 192 people without overloading.
The population of Underground riders at rush hour is almost exclusively made up of adult men and women, whose weight averages about 90 kg with an SD of about 12kg.
If the engineers who planned the Pimlico station designed the escalator to carry 10,000 kg worth of people without breaking, what proportion of the time when it is fully loaded with 192 people would it break down? (Express your answer in the form "about 1 in every k fully loaded trips.")
Given that the morning and evening peak traffic periods together last about an hour and a half, and the turnover on the escalator is such that it's like having a new trip with 192 new people about every minute during this period, do you regard this failure rate as acceptably low? (Hint: At this rate about how often would it break down?)
If the engineers had wanted the escalator to fail only about once in every 10,000 trips (which would still mean it would overload about once every 110 days), how much weight should they have designed it to carry? (Be explicit about your probability model (in other words, relate this setup explicitly to the population-and-sample framework), and comment briefly on all assumptions you make and whether you think they are reasonable.)
Explain briefly
Question? 2
In 1965, the U.S. Supreme Court decided the case of Swain vs. Alabama. Swain, a black man, was convicted in Talladega County, Alabama, of raping a white woman. He was sentenced to death. The case was appealed to the Supreme Court on the grounds that there were no blacks on the jury; moreover, no black "within the memory of persons now living has ever served on any petit jury in any civil or criminal case tried in Talladega County, Alabama."
The Supreme Court denied the appeal, on the following grounds. As provided by Alabama law, the jury was selected from a panel of about 100 persons. There were 8 blacks on the panel. (They did not serve on the jury because they were "struck," or removed, through a maneuver called peremptory challenges by the prosecution. Such challenges were until quite recently constitutionally protected.) The Supreme Court ruled that the presence of 8 blacks on the panel showed "The overall percentage disparity has been small and reflects no studied attempt to include or exclude a specified number of blacks."
At that time in Alabama, only men over the age of 21 were eligible for jury duty (!). There were 16,000 such men in Talladega County, of whom about 26% were black.
If 100 people were chosen by simple random sampling from this population, what is the chance that 8% or fewer would be black?
What do you conclude about the Supreme Court's opinion?
Question? 3
To track changes over time in the quality of education offered by primary and secondary schools in the U.S., an outfit called the National Assessment of Educational Progress (NAEP) administers tests each year in a variety of subjects to nationwide random samples of students. They actually use cluster sampling rather than simple random sampling, but their method is roughly equivalent to taking an SRS of about 950 students from the population of all U.S. students in a given grade in any given year.
In 1975 the NAEP gave a math test to 17-year-olds. One of the items on the test was as follows:
Do the following addition: 1/2 + 1/3 = ...
617 of the 950 students knew the right answer, 5/6 (the other popular response was 2/5).
Estimate p, the percentage of all 17-year-old students in the U.S. who knew the right answer, attach a give-or-take to your estimate, and construct a 95% confidence interval for p.
If they wanted to be 95% sure of getting the national percentage p right to within 2 percentage points, how many students should they have sampled?
Question? 4
Discount stores often introduce new merchandise at a special low price to induce people to try it. But in the mid-1960s a prominent psychologist predicted that in the long run this practice would actually reduce sales.
With the cooperation of a discount chain (I think it was K-Mart), an experiment was performed in 1968 to test this theory. A representative sample of 120 stores was chosen, and the stores were arranged into 60 pairs, matched according to characteristics like sales volume and location. These stores did not advertise, and displayed their merchandise in similar ways. A new kind of cookie was introduced in all 120 stores. Within each pair of stores, one was chosen at random to introduce the cookies at the special low price of 49 cents a box, with the price increasing to 69 cents after two weeks; the other store in the pair introduced the cookies at the regular price of 69 cents a box. Total sales (in cases) of the cookies were computed for each store for six weeks from the time they were introduced; the results are given below.
Does this evidence support or refute the psychologist's theory?
What was the point of pairing the stores in the way they did?
Explain briefly
Question? 5
In 1969, the well-known pediatrician Dr. Benjamin Spock came to trial before a judge named Ford in Boston's Federal court house. He was charged with conspiracy to violate the Military Service Act (in addition to his work on child development he was active in anti-war protests in the 60s). A lawyer writing about the case that same year in the Chicago Law Review said about the case, "Of all defendants at such trials, Dr. Spock, who had given wise and welcome advice on child-bearing to millions of mothers, would have liked women on his jury."
The jury was drawn from a panel of 350 persons, called a venire, selected by Judge Ford's clerk. This venire included only 102 women, even though 53% of the eligible jurors in the district were female. At the next stage in selecting the jury to hear the case, Judge Ford chose 100 potential jurors out of these 350 people. His choices included only 9 women. If 350 people are chosen from all the eligible jurors in the district, how likely is it that the sample will include 102 women or fewer?
If 100 people are chosen at random without replacement from a group of people consisting of 102 women and 248 men, what is the chance that the sample will include 9 women or fewer? (Hint: remember the correction factor, if relevant.)
What do you conclude about the impartiality of Judge Ford's selection process?
Explain briefly.
Question? 6
A rope company produces ropes on two production lines. The tensile strength is an important measure of quality. A test of randomly selected ropes yields the following results.
The company assumes the populations of rope tensile strengths are approximately normal. The company suspects there is a difference in the mean values of the populations. Conduct an appropriate hypothesis test at the 0.05 significance level.