Exam 4: Random Variables and Probability Distributions

Find a value of the standard normal random variable $z$ , called $z _ { 0 }$ , such that $P \left( z \geq z _ { 0 } \right) = 0.70$ .

A binomial random variable is defined to be the number of units sampled until x successes is observed.

Consider the given discrete probability distribution. Find the probability that $x$ exceeds 4. x 3 4 6 8 P(x) 0.25 ? 0.24 0.06

Given that $x$ is a hypergeometric random variable with $N = 10 , n = 5$ , and $r = 6$ , find each probability. a. $P ( x = 0 )$ b. $P ( x = 1 )$ c. $P ( x \leq 1 )$ d. $P ( x \geq 2 )$

According to a recent study, 1 in every 9 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability That at least 2 of the women sampled have been the victim of domestic abuse. Round to six decimal Places.

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. How many of the sampled voters do we expect to vote in the next presidential election?

Consider the given discrete probability distribution. Find P(x\leq4) . x 0 1 2 3 4 5 p(x) .30 .25 .20 .15 .05 .05

The time between arrivals at an ATM machine follows an exponential distribution with $\theta = 10$ minutes. Find the probability that more than 25 minutes will pass between arrivals.

Suppose a Poisson probability distribution with $\lambda = 12.5$ provides a good approximation of the distribution of a random variable $x . .$ Find $\mu$ for $x$ .

Given that x is a hypergeometric random variable with $N = 10 , n = 3 \text {, and } r = 6 \text {, compute } P ( x = 0 ) \text {. }$

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 440 seconds and a standard deviation of 60 seconds. Between what times do we expect approximately 95% of the boys to run the mile?

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the standard deviation of the distribution. 2 3 5 8 10 () 0.10 0.20 0.30 0.30 0.10

The expected value of a discrete random variable must be one of the values in which the random variable can result.

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

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 10$ and $d = 70$ . Find $P ( x > 55 )$ .

The printout below contains summary statistics of the heights of a sample of 200 adult men in the United States. Descriptive Statistics: HT Variable Mean StDev Minimum Q1 Median Q2 HT 200 70.187 2.716 62.375 67.875 69.625 71.500 Use the information in the printout to determine whether the distribution of heights is approximately normal. Explain your reasoning.

Classify the following random variable according to whether it is discrete or continuous. The height of a player on a basketball team

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit.

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 3000 miles. What warrantyshould the company use if they want 96% of the tires to outlast the warranty?