Chapter 11: Problem 101
In Exercises \(\mathrm{P} .100\) to \(\mathrm{P} .107,\) calculate the requested quantity. $$ 7 ! $$
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In Exercises \(\mathrm{P} .74\) to \(\mathrm{P} .77\), fill in the \(?\) to make \(p(x)\) a probability function. If not possible, say so. $$ \begin{array}{lcccc} \hline x & 1 & 2 & 3 & 4 \\ \hline p(x) & 0.1 & 0.1 & 0.2 & ? \\ \hline \end{array} $$
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.
Critical Reading on the SAT Exam In Table P.16 with Exercise \(\mathrm{P} .157\), we see that scores on the Critical Reading portion of the SAT (Scholastic Aptitude Test) exam are normally distributed with mean 495 and standard deviation \(116 .\) Use the normal distribution to answer the following questions: (a) What is the estimated percentile for a student who scores 700 on Critical Reading? (b) What is the approximate score for a student who is at the 30 th percentile for Critical Reading?
Benford's Law Frank Benford, a physicist working in the 1930 s, discovered an interesting fact about some sets of numbers. While you might expect the first digits of numbers such as street addresses or checkbook entries to be randomly distributed (each with probability \(1 / 9\) ), Benford showed that in many cases the distribution of leading digits is not random, but rather tends to have more ones, with decreasing frequencies as the digits get larger. If a random variable \(X\) records the first digit in a street address, Benford's law says the probability function for \(X\) is $$ P(X=k)=\log _{10}(1+1 / k) $$ (a) According to Benford's law, what is the probability that a leading digit of a street address is \(1 ?\) What is the probability for \(9 ?\) (b) Using this probability function, what proportion of street addresses begin with a digit greater than \(2 ?\)
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