Exam 7: The Normal Probability Distribution

Graph a Normal Curve -Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 4 and standard deviation of 1. The graphs would

State the Properties of the Normal Curve -The area under a standard normal density curve with mean of 0 and standard deviation of 1 is

Explain the Role of Area in the Normal Density Function -The analytic scores on a standardized aptitude test are known to be normally distributed with mean $\mu = 610$ and standard deviation $\sigma = 115$ . (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of test takers who scored less than 725 . (c) Suppose the area under the normal curve to the left of $X = 725$ is $0.8413$ . Provide two interpretations of this result.

Use Normal Probability Plots to Assess Normality -Determine whether the following normal probability plot indicates that the sample data could have come from a population that is normally distributed.

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

Find the Area under the Standard Normal Curve -Given a distribution that follows a standard normal curve, what does the graph of the curve do as z increases in the positive direction?

Find the Value of a Normal Random Variable -A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $600 per month and a standard deviation of $50 per month. Refer to such expenses as PCE s (personal call expenses). Find the point in the distribution below which 2.5% of the PCE s fell.

Explain the Role of Area in the Normal Density Function -The average mpg (miles per gallon) of a new model of motorcycle is known to be normally distributed with mean $\mu = 27.4 \mathrm { mpg }$ and standard deviation $\sigma = 2.9 \mathrm { mpg }$ . (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of mpgs between $29.3$ and $23.7$ . (c) Suppose the area under the normal curve to between $X = 29.3$ and $X = 23.7$ is $0.6419$ . Provide two interpretations of this result.

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 $\mathrm { n } = 10$ and $\mathrm { p } = 0.2$ .

Find the Area under the Standard Normal Curve -Find the sum of the areas under the standard normal curve to the left of $\mathrm { z } = - 1.25$ and to the right of $\mathrm { z } = 1.25$ .

Interpret the Area under the Standard Normal Curve as a Probability - $\mathrm { z } = 0.5$ and $\mathrm { z } = 1.4$

Explain the Role of Area in the Normal Density Function -The proportion of the population that has certain characteristics is the same as the probability that a randomly selected individual of the population has these same characteristics.

Approximate Binomial Probabilities Using the Normal Distribution -Assuming that all conditions are met to approximate a binomial probability distribution with the standard normal distribution, then to compute $P ( x \leq 23 )$ from the binomial distribution we must compute _____as the normal approximation.

Interpret the Area under the Standard Normal Curve as a Probability -Scores on a standardized test are normally distributed with a mean of 100 and a standard deviation of 12 . An individual's test score is found to be 125 . Find the $z$ -score corresponding to this value.

Find and Interpret the Area under a Normal Curve -Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 11.14 onces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains more than 11.14 ounces of beer.

Interpret the Area under the Standard Normal Curve as a Probability - $\mathrm { z } = - 0.2$ and $\mathrm { z } = 0.2$

Find and Interpret the Area under a Normal Curve -A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows 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). Find the probability that a randomly selected month had a PCE that falls below $550.

Find and Interpret the Area under a Normal Curve -Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.43 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains fewer than 12.33 ounces of beer.

State the Properties of the Normal Curve -Approximately _____ $\%$ of the area under the normal curve is between $\mu - 3 \sigma$ and $\mu + 3 \sigma$ .

Approximate Binomial Probabilities Using the Normal Distribution -A salesperson found that there was a 1% chance of a sale from her phone solicitations. Find the probability of getting 5 or more sales for 1000 telephone calls.