Exam 7: The Normal Probability Distribution

Find z-scores for a Given Area -For a standard normal curve, find the z-score that separates the bottom $30 \%$ from the top $70 \%$ .

Approximate Binomial Probabilities Using the Normal Distribution -A local rental car agency has 100 cars. The rental rate for the winter months is 60%. Find the probability that in a given winter month fewer than 70 cars will be rented. Use the normal distribution to approximate the binomial distribution.

Provide an appropriate response -A random number generator is set top generate integer random numbers between 1 and 10 inclusive following a uniform distribution. What is the probability of the random number generator generating a 7

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. - $X = 60 ; \sigma = 7$

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = 0$ and $\mathrm { z } = 3$ .

Approximate Binomial Probabilities Using the Normal Distribution -A local concert center reports that it has been experiencing a 15% rate of no-shows on advanced reservations. Among 150 advanced reservations, find the probability that there will be fewer than 20 no-shows.

Find and Interpret the Area under a Normal Curve -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 equal to 440 tomatoes per day and a standard deviation equal to 30 tomatoes per day. If there are 398 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

A random variable X is normally distributed with = 60. Convert the value of X to a z-score, if the standard deviation is as given. -Suppose a random variable $x$ is normally distributed with mean, $\mu$ , and standard deviation, $\sigma$ . Then the random variable $z = \frac { \bar { x } - \mu } { \sigma }$ has the graph of the

Find the Value of a Normal Random Variable -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 equal to 133 tomatoes per day and a standard deviation equal to 30 tomatoes per day. 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

Find the Value of a Normal Random Variable -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 2900 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty

Explain the Role of Area in the Normal Density Function -The weight of 2-year old hyraxes is known to be normally distributed with mean $\mu = 2200$ grams and standard deviation $\sigma = 365$ grams (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of hyraxes who weighed more than 2930 grams. (c) Suppose the area under the normal curve to the left of $X = 2930$ is $0.0228$ . Provide two interpretations of this result.

Approximate Binomial Probabilities Using the Normal Distribution -The author of an economics book has trouble deciding whether to use the words he or she in the book s examples. To solve the problem, the author flips a coin each time the problem arises. If a head shows, the author uses he and if a tail shows, the author uses she . If this problem occurs 100 times in the book, what is the probability that she will be used 58 times

Approximate Binomial Probabilities Using the Normal Distribution -For the following conditions, determine if it is appropriate to use the normal distribution to approximate a binomial distribution with $\mathrm { n } = 48$ and $\mathrm { p } = 0.3$ .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = 1$ and $\mathrm { z } = 2$ .

Interpret the Area under the Standard Normal Curve as a Probability - $\mathrm { z } = - 2$ and $\mathrm { z } = - 0.1$

Explain the Role of Area in the Normal Density Function -The area under the normal curve drawn with regard to the population parameters is the same as the probability that a randomly selected individual of a population has these characteristics.

Find and Interpret the Area under a Normal Curve -Use the standard normal distribution to find $\mathrm { P } ( \mathrm { z } < - 2.33$ or $\mathrm { z } > 2.33 )$ .

Find and Interpret the Area under a Normal Curve -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 460 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 322 seconds.

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve between $\mathrm { z } = 1.5$ and $\mathrm { z } = 2.5$ .

Use Normal Probability Plots to Assess Normality -The following data represent a random sample of the number of shares of a pharmaceutical company's stock traded for 20 days in 2000 . 4.54 9.46 10.96 8.08 14.4 5.85 10.25 9.5 12.66 13.81 4.8 9.1 12.15 7.31 28.1 8.27 10.7 4.84 7.8 11.44