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Name: _______________________________________________ Per: ____ Date: ______________Chapter 6 Problem Set A.P. Statistics1) A four sided die, shaped like an asymmetrical tetrahedron, has the following roll probabilities. Number on Die1234Probability0.40.30.20.1 Let X = the result of a single roll. (a) Find P(1<X<4) (b) Find P(X ≠ 3) (c) Describe P (x = 3 / x ≥ 2) in words and find its value. (d) Find the smallest value A for which P(X<A) > 0.6 (e) If T = the sum of two rolls, find P(T = 4) (f) Find and interpret the mean and standard deviation of X.2) As of December 2008, various polls indicate that 35% of people who use the internet have profiles on at least one social networking site. We will discover later in the text that if you take a sample of 50 internet users, the proportion of the sample who have profiles on social networking sites can be considered a random variable. Moreover, assuming the 35% is accurate, this random variable will be approximately Normally distributed with a mean of 0.35 and a standard deviation of 0.067. What is the probability that the proportion of a sample of size 50 who have profiles on social networking sites is greater than 0.5? 3) The rules for means and variances allow you to find the mean and variance of a sum of random variables without first finding the distribution of the sum, which is usually much harder to do.(a) A single toss of a balanced coin results in either 0 or 1 head, each with probability 1/2. What are the mean and standard deviation of the number of heads?(b) Toss a coin four times. Use the rules for means and variances to find the mean and standard deviation of the total number of heads.4) You have two instruments with which to measure the height of a tower. If the true height is 100 meters, measurements with the first instrument vary with mean 100 meters and standard deviation 1.2 meters. Measurements with the second instrument vary with mean 100 meters and standard deviation 0.65 meters. You make one measurement with each instrument. Your results are X1 for the first and X2 for the second and the measurements are independent.(a) To combine the two measurements, you might average them, Y=12(X1+X2). What are the mean and standard deviation of Y?(b) It makes sense to give more weight to the less variable measurement because it is more likely to be closer to the truth. Statistical theory says that to make the standard deviation as small as possible you should weight the two measurements inversely proportional to their variances. The variance of X2 is very close to half the variance of X1, so X2 should get twice the weight of X1. That is, use W=13X1+23X2. What are the mean and standard deviation of W?5) Lamar and Lawrence run a two-person lawn-care service. They have been caring for Mr.Johnson’s very large lawn for several years, and they have found that the time it takes Lamarto mow the lawn itself is approximately Normally distributed with a mean of 105 minutesand a standard deviation of 10 minutes. Meanwhile, the time it takes for Lawrence to use theedger and string trimmer to attend to details is also Normally distributed with a mean of 98minutes and a standard deviation of 15 minutes. They prefer to finish their jobs within 5minutes of each other. What is the probability that this happens, assuming their finish timesare independent?6) Determine whether each random variable described below satisfies the conditions for abinomial setting, a geometric setting, or neither. Support your conclusion in each case.(a) Draw a card from a standard deck of 52 playing cards, observe the card, return the card tothe deck, and shuffle. Count the number of times you draw a card in this manner untilyou observe a jack.(b) Joey buys a Virginia lottery ticket every week. X is the number of times in a year that hewins a prize.7) When a computerized generator is used to generate random digits, the probability that anyparticular digit in the set {0, 1, 2, . . . , 9} is generated on any individual trial is 1/10 = 0.1.Suppose that we are generating digits one at a time and are interested in tracking occurrencesof the digit 0.(a) Determine the probability that the first 0 occurs as the fifth random digit generated.(b) How many random digits would you expect to have to generate in order to observe thefirst 0?(c) Let X = number of digits selected until first zero is encountered. Construct a probability distribution histogram for X = 1 through X = 5. 8) Suppose there are 1100 students in your high school, and 28% of them take Spanish. Youselect a sample of 50 student in the school, and you want to calculate the probability that 15or more of the students in your sample take Spanish. Which condition for the binomialsetting has been violated here, and why does the binomial distribution do a good job ofestimating this probability anyway?9) A fair coin is flipped 20 times.(a) Determine the probability that the coin comes up tails exactly 15 times.(b) Let X = the number of tails in the 20 flips. Find the mean and standard deviation of X.(c) Find the probability that X takes a value within 1 standard deviation of its mean. ................
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