Exam 5: Continuous Random Variables

The preventable monthly loss at a company has a normal distribution with a mean of $5700 and a standard deviation of $30. A new policy was put into place, and the preventable loss the next Month was $5520. What inference can you make about the new policy?

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 8600 miles. If the manufacturer guarantees the tread life of the tires for the first 49,680 miles, what proportion of the tires will need to be replaced under warranty?

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one Tattoo. Find the approximate probability that more than 30 of the sampled students had at least one Tattoo.

Assume that x is a binomial random variable with n = 100 and p = 0.60. Use a normal approximation to find $P ( x \leq 55 )$

The length of time (in months)that a cashier works for a certain fast food restaurant is exponentially distributed with a mean of 7 months. a. Find the probability that a cashier works for the restaurant for at least 2 years. b. Find the probability that a cashier works for the restaurant for less than 1 month.

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one Tattoo. Find the mean and standard deviation for this binomial distribution.

Suppose a random variable $x$ is best described by a normal distribution with $\mu = 60$ and $\sigma = 3$ . Find the $z$ -score that corresponds to the value $x = 63$ .

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 20 to 70. What is the probability that this experiment results in an outcome Less than 30? Round to the nearest hundredth when necessary.

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

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 582 tomatoes and a standard deviation of 30 tomatoes. If there are 540 tomatoes available to be sold at the roadside stand at the beginning of a day, what is the probability that they will all be sold?

The following data represent the scores of a sample of 50 students on a statistics exam. The mean score is $\bar { x } = 80.3$ , and the standard deviation is $s = 11.37$ . 49 51 59 63 66 68 68 69 70 71 71 71 73 74 76 76 76 77 78 79 79 79 79 80 80 82 83 83 83 85 85 86 86 88 88 88 88 89 89 89 90 91 92 92 93 95 96 97 97 98 What percentage of the scores fall in each of the intervals $\bar { x } \pm s , \bar { x } \pm 2 s$ , and $\bar { x } \pm 3 s$ ? Based on these percentages, do you believe that the distribution of scores is approximately normal? Explain.

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

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

The board of examiners that administers the real estate brokerʹs examination in a certain state found that the mean score on the test was 507 and the standard deviation was 72. If the board wants to set the passing score so that only the best 10% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

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.

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

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $500 per month and a standard deviation of $50 per Month. Refer to such expenses as PCEʹs (personal call expenses). Using the distribution above, What is the probability that during a randomly selected month PCEʹs were between $375.00 and $590.00?

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door Exceeds 50 years old. Round to the nearest ten-thousandth.

Suppose that $x$ has an exponential distribution with $\theta = 1.75$ . Find each probability. a. $P ( x \geq 2 )$ b. $P ( x < 2 )$

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one Tattoo. Find the approximate probability that more than 17 and less than 26 of the sampled Students had at least one tattoo.