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Chapter 7: Problem 66

Let \(z\) denote a random variable that has a standard normal distribution. Determine each of the following probabilities: a. \(P(z<2.36)\) b. \(P(z \leq 2.36)\) c. \(P(z<-1.23)\) d. \(P(1.142)\) g. \(P(z \geq-3.38)\) h. \(P(z<4.98)\)

Short Answer

Expert verified
a. 0.9906, b. 0.9906, c. 0.1093, d. 0.0101, e. 0.0588, f. 0.0228, g. 0.9996, h. 1

Step by step solution

01

Understanding Standard Normal Distribution Table

A standard normal distribution table, or Z-table, is used to find the probability that a statistic is less than a given Z-score. This table gives the area (probability) that lies to the left of a given Z-score.
02

Finding the probabilities

Using the Z-table, find the probabilities for each part of the exercise. Here are the probabilities for each part: a. \(P(z<2.36)\) = 0.9906b. \(P(z \leq 2.36)\) = 0.9906 (Whether the inequality is less than or less than and equal to does not matter in this context because we're dealing with a continuous distribution.)c. \(P(z<-1.23)\) = 0.1093d. \(P(1.142)\) = 1 - 0.9772 = 0.0228 (for z-scores greater than a number, we subtract the z-table value from 1)g. \(P(z \geq -3.38)\) = 0.9996h. \(P(z < 4.98)\) = essentially 1 or 100%, as the Z-table typically does not go past 3.49, and anything further is so close to 0 or 1 that it is typically recorded as such.

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Most popular questions from this chapter

A local television station sells \(15-\mathrm{sec}, 30-\mathrm{sec}\), and 60 -sec advertising spots. Let \(x\) denote the length of a randomly selected commercial appearing on this station, and suppose that the probability distribution of \(x\) is given by the following table: $$ \begin{array}{lrrr} x & 15 & 30 & 60 \\ p(x) & .1 & .3 & .6 \end{array} $$ a. Find the average length for commercials appearing on this station. b. If a 15 -sec spot sells for $$\$ 500$$, a 30 -sec spot for $$\$ 800$$, and a 60 -sec spot for $$\$ 1000$$, find the average amount paid for commercials appearing on this station. (Hint: Consider a new variable, \(y=\) cost, and then find the probability distribution and mean value of \(y .\) )

An experiment was conducted to investigate whether a graphologist (a handwriting analyst) could distinguish a normal person's handwriting from that of a psychotic. A well-known expert was given 10 files, each containing handwriting samples from a normal person and from a person diagnosed as psychotic, and asked to identify the psychotic's handwriting. The graphologist made correct identifications in 6 of the 10 trials (data taken from Statistics in the Real World, by R. J. Larsen and D. F. Stroup [New York: Macmillan, 1976\(]\) ). Does this evidence indicate that the graphologist has an ability to distinguish the handwriting of psychotics? (Hint: What is the probability of correctly guessing 6 or more times out of 10 ? Your answer should depend on whether this probability is relatively small or relatively large.)

The Wall Street Journal (February 15,1972 ) reported that General Electric was sued in Texas for sex discrimination over a minimum height requirement of \(5 \mathrm{ft}\) 7 in. The suit claimed that this restriction eliminated more than \(94 \%\) of adult females from consideration. Let \(x\) represent the height of a randomly selected adult woman. Suppose that \(x\) is approximately normally distributed with mean 66 in. (5 ft 6 in.) and standard deviation 2 in. a. Is the claim that \(94 \%\) of all women are shorter than \(5 \mathrm{ft}\) 7 in. correct? b. What proportion of adult women would be excluded from employment as a result of the height restriction?

Consider a large ferry that can accommodate cars and buses. The toll for cars is $$\$ 3$$, and the toll for buses is $$\$ 10 .$$ Let \(x\) and \(y\) denote the number of cars and buses, respectively, carried on a single trip. Cars and buses are accommodated on different levels of the ferry, so the number of buses accommodated on any trip is independent of the number of cars on the trip. Suppose that \(x\) and \(y\) have the following probability distributions: $$ \begin{array}{lrrrrrr} x & 0 & 1 & 2 & 3 & 4 & 5 \\ p(x) & .05 & .10 & .25 & .30 & .20 & .10 \\ y & 0 & 1 & 2 & & & \\ p(y) & .50 & .30 & .20 & & & \end{array} $$ a. Compute the mean and standard deviation of \(x\). b. Compute the mean and standard deviation of \(y\). c. Compute the mean and variance of the total amount of money collected in tolls from cars. d. Compute the mean and variance of the total amount of money collected in tolls from buses. e. Compute the mean and variance of \(z=\) total number of vehicles (cars and buses) on the ferry. f. Compute the mean and variance of \(w=\) total amount of money collected in tolls.

A gas station sells gasoline at the following prices (in cents per gallon, depending on the type of gas and service): \(315.9,318.9,329.9,339.9,344.9\), and 359.7. Let \(y\) denote the price per gallon paid by a randomly selected customer. a. Is \(y\) a discrete random variable? Explain. b. Suppose that the probability distribution of \(y\) is as follows: $$ \begin{array}{lrrrrrr} y & 315.9 & 318.9 & 329.9 & 339.9 & 344.9 & 359.7 \\ p(y) & .36 & .24 & .10 & .16 & .08 & .06 \end{array} $$ What is the probability that a randomly selected customer has paid more than $$\$ 3,20$$ per gallon? Less than $$\$ 3.40$$ per gallon? c. Refer to Part (b), and calculate the mean value and standard deviation of \(y .\) Interpret these values.
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