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Exam 5: Continuous Random Variables

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Question 71

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The number of children in a family can be modelled using a continuous random variable.

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Question 72

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Determine the value of $e ^ { - a / \theta } \text { for } \theta = 2 \text { and } a = 3$

A) 0.223130
B) 0.513417
C) -4.481689
D) 4.481689

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Question 73

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Use the standard normal distribution to find $\mathrm { P } ( \mathrm { z } < - 2.33 \text { or } \mathrm { z } > 2.33 )$

Suppose a random variable x is best described by a normal distribution with $\mu = 60 \text { and } \sigma = 5 \text {. }$ . Find the z-score that corresponds to the value x = 65.

The weight of corn chips dispensed into a 24-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 24.5 ounces and a standard deviation of 0.2 ounce. What proportion of the 24-ounce bags contain more than the advertised 24 ounces of chips?

It is against the law to discriminate against job applicants because of race, religion, sex, or age. Of the individuals who apply for an accountant's position in a large corporation, 39% are over 45 years old. If the company decides to choose 96 of a very large number of applicants for closer credential screening, claiming that the selection will be random and not age-biased, what is the z-value associated with fewer than 46 of those chosen being over 45 years old? (Assume that the applicant pool is large enough so that x, the number in the sample over 45 years old, has a binomial probability distribution.)

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 1700 miles. What is the probability a certain tire of this brand will last between 56,430 miles and 56,940 miles?

For any continuous probability distribution, P(x = c) = 0 for all values of c.

The rate of return for an investment can be described by a normal distribution with mean 47% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 53%?

Suppose x is a random variable best described by a uniform probability distribution with c = 60 and d = 20. Find P(x ≥ 60) .

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability that between 538 and 546 drivers in a sample of 700 drivers wear seat belts?

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 2300 miles. What is the probability a particular tire of this brand will last longer than 57,700 miles?

Assume that x is a binomial random variable with n = 1000 and p = 0.80. Use a normal approximation to find P(800 < x ≤ 820) .

A statistician received some data to analyze. The sender of the data suggested that the data was normally distributed. Which of the following methods can be used to determine if the data is, in fact, normally distributed? I. Construct a histogram and/or stem-and-leaf display of the data and check the shape. II. Compute the intervals $\bar { x } \pm s , \bar { x } \pm 2 s$ , and $\bar { x } \pm 3 s$ , and determine the percentage of measurements falling in each. Compare these percentages to $68 \% , 95 \%$ , and $100 \%$ . III. Calculate a value of $\frac { \mathrm { IQR } } { \mathrm { s } }$ . If this value is approximately $1.3$ , then the data is normal. IV. Construct a normal probability plot of the data. If the points fall on a straight line, then the data is normal.

Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 124 tomatoes and a standard deviation of 30 tomatoes. How many tomatoes must be available on any given day so that there is only a 1.5% chance that all tomatoes will be sold?

Suppose x is a uniform random variable with c = 10 and d = 80. Find the mean of the random variable x.

When the points on a normal probability plot lie approximately on a straight line, the data are approximately normally distributed.

The exponential distribution has the property that its mean equals its standard deviation.

Use the standard normal distribution to find $P ( - 2.50 < z < 1.50 )$