Cover Page Homework #8

Size: px

Start display at page:

Download "Cover Page Homework #8"

Katherine Harrell
5 years ago
Views:

1 MODESTO JUNIOR COLLEGE Department of Mathematics MATH 134 Fall 2011 Problem 11.6 Cover Page Homework #8 (a) What does the population distribution describe? (b) What does the sampling distribution of x describe? The sampling distribution is the Imaginary Data set that we ve discussed in class. Problem (b) Answer: Problem (b) Answer: Problem (a) P (3 out of 3) (b) P ( 2 out of 3)

2 3. (a) Chance error: (b) Relative chance error: 4. True or False? Explain. 5. Explain your answers within the attached assignment. (a) 10 or 100? (b) 10 or 100? (c) 10 or 100? 6. Which is better? (a) (i) or (ii)? 7. Yes or No? Explain. Attach problems #9, #10 and #11 to the end of your homework packet. In each of these problems, use the Probability Model or Statistical Model with the Population, Sample and Imaginary Data set clearly defined and illustrated along with the corresponding histograms, as shown in class. 2

3 Homework #8 Due at the beginning of class on Tuesday, November 1 st Note: In each of the following problems, use the Probability Model with the Population, Sample and Imaginary Data clearly defined, illustrated, along with corresponding histograms to solve the problem, as shown in class. 1. Problems from Moore s text: 11.27, 11.28, 11.29, 11.30, Given that the chance I make a free throw is 6 out of 10, what is the probability that I make 3 free throws in a row? What is the probability that I make at least 2 out of 3 free throws? 3. A machine has been designed to toss a coin automatically and keep track of the number of heads. After 1,000 tosses, it has 550 heads. Express the chance error both in absolute terms and as a percentage of the number of tosses. 4. A coin is tossed 100 times, landing heads 53 times. However, the last seven tosses are all heads. True or false: the chance that the next toss will be heads is somewhat less than 50%. Explain. 5. (a) A coin is tossed, and you win a dollar if there are more than 60% heads. Which is better: 10 tosses or 100? Explain. (b) As in (a), but you win the dollar if there are between 40% and 60% heads. (c) As in (a), but you win the dollar if there are exactly 50% heads. 6. With a Nevada roulette wheel, there are 18 chances in 38 that the ball will land in a red pocket. A wheel is going to be spun many times. There are two choices: (i) 38 spins, and you win a dollar if the ball lands in a red pocket 20 or more times. (ii) 76 spins, and you win a dollar if the ball lands in a red pocket 40 or more times. Which is better? Or are they the same? Explain. 7. Do the following problems from Moore s text: Problems: 11.5, In the winter of 1976, nineteen students in an introductory statistics class at the University of California, Berkeley, were asked to measure the thickness of a table top, using a vernier gauge reading to one-thousandth of an inch. Each person made two measurements, shown in the table below (in inches). For example, the first person got inches on her/his first measurement and inches on the second. 3

4 Measurement (inches) Person First Second (a) Did all of the students work independently of one another? Explain briefly. 9. At Nevada roulette tables, the house special is a bet on the numbers {0, 00, 1, 2, 3} The bet pays 6 to 1 (e.g., you can either lose $1 or gain +$6 on a $1 bet.) There are 5 chances in 38 to win. Find the expected Gain or Loss in 1000 plays of roulette when betting $1 on the house special by working through the following steps: (a) How much money do you expect to gain or lose on a single $1 bet on the house special? Draw out the population, sample and imaginary data sets, as we ve done in class in order to answer parts (a), (b) and (c). (Hint: Consider the population of gains/losses for the possible outcomes of a single spin of a roulette wheel.) (b) Use the Math Fact to find the standard deviation, σ, for the average gain or loss on a single spin of the roulette wheel. (c) Draw the histograms for the population and sample. 4

5 (d) In 1000 plays of roulette when betting $1 on the house special, the expected gain or loss is about $, give or take $ or so. (Find the expected gain/loss and the standard error of this estimate.) (e) After 1000 spins of the roulette wheel, betting $1 on the house special each time, estimate the chance of coming out ahead. As part of your answer, draw the long run histogram of Sums with the E IID (S) and SE IID (S) indicated. 10. (business) Investors remember 1987 as the year stocks lost 22% of their value in a single day. For 1987 as a whole, the mean return of all common stocks on the New York Stock Exchange was µ = 3.5%. (That is, these stocks lost an average of 3.5% of their value in 1987.) The standard deviation of the returns was about σ = 26%. The distribution of annual returns for stocks is roughly normal. (a) What percent of stocks lost money? (That is the same as the probability that a stock chosen at random has a return less than 0.) (b) Suppose that you held a portfolio of 5 stocks chosen at random from New York Stock Exchange stocks. What are the mean and standard deviation of the returns of randomly chosen portfolios of 5 stocks? (c) What percent of such portfolios lost money? Explain the difference between this result and the result of (a). 11. (business) As we saw in the C&O freight study in class, it has become common in recent years for evidence based on sampling to be used informally by companies, or even formally presented in court, to figure out accounts receivable or settle disputes. For instance, when sample evidence is submitted to a court to justify deductions from taxable profits for expenses incurred in a long series of transactions, it is typically stipulated by the court that the sample estimate of the average of the average expenses per transaction be correct to within some tolerance like $500 with at least 95% probability. From past experience a given firm will know how much variability the distribution of expenses per transaction has, for example a standard deviation of $1800. The company you work for decides tentatively to take a simple random sample of 40 transactions to estimate the overall average expenses per transaction. Create a statistical model, describe the population, sample and imaginary datasets. Identify the variable of interest. Draw the shape of the population, sample and imaginary data set histograms. Show that the chance your sample mean will be correct to within the court s stipulation is only about 92%. How many observations would you actually need to satisfy the court? Explain briefly.

Part 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?

Part 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going? 1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard