Chapter 11: Problem 108
In Exercises \(\mathrm{P} .108\) to \(\mathrm{P} .111,\) calculate the requested binomial probability. Find \(P(X=2)\) if \(X\) is a binomial random variable with \(n=6\) and \(p=0.3\).
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Random Samples of College Degree Proportions In Exercise P.162, we see that the distribution of sample proportions of US adults with a college degree for random samples of size \(n=500\) is \(N(0.325,0.021) .\) How often will such samples have a proportion, \(\hat{p},\) that is more than \(0.35 ?\)
Ask you to convert an area from one normal distribution to an equivalent area for a different normal distribution. Draw sketches of both normal distributions, find and label the endpoints, and shade the regions on both curves. The lower \(10 \%\) for a standard normal distribution converted to a \(N(500,80)\) distribution
The Standard and Poor 500 (S\&P 500 ) is a weighted average of the stocks for 500 large companies in the United States. It is commonly used as a measure of the overall performance of the US stock market. Between January 1,2009 and January \(1,2012,\) the S\&P 500 increased for 423 of the 756 days that the stock market was open. We will investigate whether changes to the S\&P 500 are independent from day to day. This is important, because if changes are not independent, we should be able to use the performance on the current day to help predict performance on the next day. (a) What is the probability that the S\&P 500 increased on a randomly selected market day between January 1,2009 and January \(1,2012 ?\) (b) If we assume that daily changes to the \(S \& P\) 500 are independent, what is the probability that the S\&P 500 increases for two consecutive days? What is the probability that the S\&P 500 increases on a day, given that it increased the day before? (c) Between January 1, 2009 and January 1,2012 the S\&P 500 increased on two consecutive market days 234 times out of a possible \(755 .\) Based on this information, what is the probability that the S\&P 500 increases for two consecutive days? What is the probability that the S\&P 500 increases on a day, given that it increased the day before? d) Compare your answers to part (b) and part (c). Do you think that this analysis proves that daily changes to the S\&P 500 are not independent?
Class Year Suppose that undergraduate students at a university are equally divided between the four class years (first-year, sophomore, junior, senior) so that the probability of a randomly chosen student being in any one of the years is \(0.25 .\) If we randomly select four students, give the probability function for each value of the random variable \(X=\) the number of seniors in the four students.
Airline Overbooking Suppose that past experience shows that about \(10 \%\) of passengers who are scheduled to take a particular flight fail to show up. For this reason, airlines sometimes overbook flights, selling more tickets than they have seats, with the expectation that they will have some no shows. Suppose an airline uses a small jet with seating for 30 passengers on a regional route and assume that passengers are independent of each other in whether they show up for the flight. Suppose that the airline consistently sells 32 tickets for every one of these flights. (a) On average, how many passengers will be on each flight? (b) How often will they have enough seats for all of the passengers who show up for the flight?
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