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Chapter 6: Q. 23 (page 356)

23. ITBS scores The Normal distribution with mean 渭=6.8and standard deviation 蟽=1.6is a good description of the Iowa Test of Basic Skills (ITBS) vocabulary scores of seventh-grade students in Gary, Indiana. Call the score of a randomly chosen student Xfor short. Find P(X鈮�9)and interpret the result. Follow the four-step process.

Short Answer

Expert verified

The probability for P(X鈮�9)is 0.0838. There is 8.38%chances that student's score is 9or more.

Step by step solution

01

Given information

The Normal distribution with mean 渭=6.8and a standard deviation of 蟽=1.6.

02

Explanation

Let Xbe a random variable with a mean of 6.8and a standard deviation of 1.6and a normal distribution.

Thez-score is calculated by dividing the value reduced by the mean by the standard deviation:
z=x-渭蟽
Where, xindicates raw score, 渭indicates population mean, and蟽 indicates the population standard deviation.
P(X鈮�9)can be calculated as:

P(x鈮�9)=Px鈭�渭蟽鈮�9鈭�渭蟽=Pz鈮�9鈭�6.81.6=P(Z鈮�1.38)=0.0838

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Most popular questions from this chapter

Lie detectors A federal report finds that lie detector tests given to truthful persons have probability about 0.2 of suggesting that the person is deceptive. A company asks 12job applicants about thefts from
previous employers, using a lie detector to assess their truthfulness. Suppose that all 12 answer truthfully. Let X=the number of people who the lie detector says are being deceptive.
(a) Find and interpret 渭X.
(b) Find and interpret 蟽X.

Ana is a dedicated Skee Ballplayer (see photo) who always rolls for the 50-point slot. The probability distribution of Ana's score Xon a single roll of the ball is shown below. You can check that 渭X=23.8and 蟽X=12.63.

(a) A player receives one ticket from the game for every 10points scored. Make a graph of the probability distribution for the random variable T=number of tickets Ana gets on a randomly selected throw. Describe its shape.

(b) Find and interpret 渭T.

(c) Compute and interpret 蟽T.

18. Life insurance

(a) It would be quite risky for you to insure the life of a 21-year-old friend under the terms of Exercise 14. There is a high probability that your friend would live and you would gain \(1250in premiums. But if he were to die, you would lose almost \)100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 21-year-old men.

(b) The risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is, the riskier the
investment. We can measure the great risk of insuring a single person鈥檚 life in Exercise 14by computing the standard deviation of the income Y that the insurer will receive. Find 蟽Y using the distribution and mean found in Exercise 14.

51. His and her earnings A study of working couples measures the income Xof the husband and the income Yof the wife in a large number of couples in which both partners are employed. Suppose that you knew the means 渭Xand 渭Yand the variances 蟽X2and 蟽纬2 of both variables in the population.
(a) Is it reasonable to take the mean of the total income X+Yto be 渭X+渭Y ? Explain your answer.
(b) Is it reasonable to take the variance of the total income to be 蟽X2+蟽y2 ? Explain your answer.

85. Aircraft engines Engineers define reliability as the probability that an item will perform its function under specific conditions for a specific period of time. A certain model of aircraft engine is designed so that each engine has probability 0.999 of performing properly for an hour of flight. Company engineers test an SRS of 350 engines of this model. Let X= the number that operate for an hour without failure.
(a) Explain whyX is a binomial random variable.

(b) Find the mean and standard deviation ofX. Interpret each value in context.
(c) Two engines failed the test. Are you convinced that this model of engine is less reliable than it鈥檚 supposed to be? Compute P(X鈮�348) and use the result to justify your answer.
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