Exam 5: Continuous Random Variables

The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a standard deviation of 0.16 ounce. Every can that has more than 12.40 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the mean amount of soda the machine should dispense if the company wants to limit the percentage that must be cleaned because of spillage to 3%?

Data has been collected and a normal probability plot for one of the variables is shown below. Based on your knowledge of normal probability plots, do you believe the variable in question is normally distributed? The data are represented by the"o" symbols in the plot.

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 30 to 80. What is the mean outcome of this experiment?

Find a value of the standard normal random variable z, called z0, such that P(z ≤ z0) = 0.70.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

A machine is set to pump cleanser into a process at the rate of 9 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 8.5 to 9.5 gallons per minute. Find the probability that the machine pumps less than 8.75 gallons during a randomly selected minute.

The time between equipment failures (in days) at a particular factory is exponentially distributed with a mean of 4.5 days. A machine just failed and was repaired today. a. Find the probability that another machine will fail within the next day. b. Find the probability that there will be no more equipment failures in the next week.

Suppose a random variable x is best described by a normal distribution with $\mu = 60 \text { and } \sigma = 13 \text {. }$ . Find the z-score that corresponds to the value x = 112.

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.

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $700 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Using the distribution above, what is the probability that during a randomly selected month PCE's were between $575.00 and $790.00?

Use the standard normal distribution to find $P ( - 2.25 < z < 1.25 )$

Suppose a random variable x is best described by a normal distribution with μ = 60 and σ = 12. Find the z-score that corresponds to the value x = 60.

The rate of return for an investment can be described by a normal distribution with mean 27% and standard deviation 3%. What is the probability that the rate of return for the investment will be at least 22.5%?

A loan officer has 66 loan applications to screen during the next week. If past record indicates that she turns down 17% of the applicants, what is the z-value associated with 61 or more of the 66 applications being rejected?

Use the standard normal distribution to find $P ( 0 < z < 2.25 )$

Which of the following is not a method used for determining whether data are from an approximately normal distribution? A) Construct a histogram or stem-and-leaf display. The shape of the graph or display should be uniform (evenly distributed). B) Compute the intervals $\bar { x } \pm s , \bar { x } \pm 2 s$ , and $\bar { x } \pm 3$ . The percentages of measurements falling in each should be approximately $68 \% , 95 \%$ , and $100 \%$ respectively. C) Find the interquartile range, $\mathrm { IQR }$ , and standard deviation, $s$ , for the sample. Then $\frac { \mathrm { IQR } } { \mathrm { s } } \approx 1.3$ . D) Construct a normal probability plot. The points should fall approximately on a straight line.

The continuity correction factor is the name given to the .5 adjustment necessary when estimating the binomial with the normal distribution.

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 80. What is the probability that this experiment results in an outcome less than 50? Round to the nearest hundredth when necessary.

Suppose that x has an exponential distribution with $\theta = 1.5 \text {. Find } P ( x > 1 )$

The board of examiners that administers the real estate broker's examination in a certain state found that the mean score on the test was 435 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.