1) Consider the given discrete probability distribution. Find P(x 3). X 1 2 3 4 5 P(x) 0.1 0.2 0.2 0.3 0.2 2) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the mean and standard deviation of the distribution. X 2 3 5 8 10 P(x) 0.1 0.2 0.3 0.3 0.1 3) An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 14 of these cars are independently sampled, what is the probability that more than half need new gas tanks? 4) According to a published study, 1 in every 4 men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled twenty - five men and asked each whether he has been involved in a minor traffic accident. How many of the 25 men do we expect to have never been involved in a minor traffic accident? Round to the nearest whole number. 5) A small life insurance company has determined that on the average it receives 3 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.6) A machine is set to pump cleanser into a process at the rate of 9 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 8.5 to 9.5 gallons per minute. Find the probability that the machine pumps less than 8.75 gallons during a randomly selected minute. 7) A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 7.5 gallons per minute. Find the variance of the distribution. 8) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2600 miles. If the manufacturer guarantees the tread life of the tires for the first 56,880 miles, what proportion of the tires will need to be replaced under warranty? 9) Which one of the following suggests that the data set is not approximately normal? Provide a brief description on each of A, B, C and D. a. A data set with 68% of the measurements within X i 23. b. The following probability plot Normal 0.9? Probability 15 99.09999 P 3 _ \"9;...on u u m c. The following Stem and Leaves plot Stem-and-Ieaf of Delta N = 27 3 3 339 '3-'188999 $55675 ' 15 w ,. \\J 0') '4 on U! J' Lsc.' urn-t = F d. A data set with IQR = 752 and s = 574. 10) You are performing a study about the weight of preschoolers. A previous study found the weights to be normally distributed with a mean of 30 pounds and a standard deviation of 4 pounds. You randomly sample 30 preschool children and find their weights (in pounds) to be as follows. Use Minitab to draw a histogram with t to display the data. Is it reasonable to assume that the weights are normally distributed? Why? 25 25 26 26.5 27 27 27.5 28 28 28.5 29 29 30 30 30.5 31 31 32 32.5 32.5 33 33 34 34.5 35 35 37 37 38 38