Exam 7: The Normal Probability Distribution

Approximate Binomial Probabilities Using the Normal Distribution -In order to use a normal approximation to the binomial probability distribution, np(1 - p) 10.

Interpret the Area under the Standard Normal Curve as a Probability -Use the standard normal distribution to find $\mathrm { P } ( - 2.25 < \mathrm { Z } < 1.25 )$ .

Interpret the Area under the Standard Normal Curve as a Probability -Use the standard normal distribution to find $\mathrm { P } ( \mathrm { Z } < - 2.33$ or $\mathrm { Z } > 2.33 )$ .

Approximate Binomial Probabilities Using the Normal Distribution -A recent survey found that 72% of all adults over 50 own cell phones. You randomly select 45 adults over 50, and ask if he or she owns a cell phone. Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the mean and standard deviation, If not, explain why.

Find z-scores for a Given Area -Find the $z$ -score that is greater than the mean and for which $70 \%$ of the distribution's area lies to its left.

Interpret the Area under the Standard Normal Curve as a Probability -Use the standard normal distribution to find $\mathrm { P } ( - 2.25 < Z < 0 )$ .

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the right of $\mathrm { z } = 1$ .

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 followed a normal distribution with an average of $800 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 a randomly selected month had a PCE of between $675.00 and $890.00

Approximate Binomial Probabilities Using the Normal Distribution -If the probability of a newborn kitten being female is 0.5, find the probability that in 100 births, 55 or more will be female. Use the normal distribution to approximate the binomial distribution.

Find the Value of a Normal Random Variable -The amount of corn chips dispensed into a 12-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 12.5 ounces and a standard deviation of 0.2 ounce. What chip amount represents the 67th percentile for the bag weight distribution

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. - $X = 50 ; \sigma = 10$

Find the Area under the Standard Normal Curve -Find the area under the standard normal curve to the left of $\mathrm { z } = 1.5$ .

State the Properties of the Normal Curve -The highest point on the graph of the normal density curve is located at

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. - $X = 45 ; \sigma = 4$

Provide an appropriate response -A machine is set to pump cleanser into a process at the rate of 5 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 4.5 to 7.5 gallons per minute. Find the probability that between 5.0 gallons and 6.0 gallons are pumped during a randomly selected minute.

Find the Value of a Normal Random Variable -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 12.15 ounces and a standard deviation of 0.04 ounce. The company is interested in reducing the amount of extra beer that is poured into the 12 ounce bottles. The company is seeking to identify the highest 1.5% of the fill amounts poured by this machine. For what fill amount are they searching

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 0 and standard deviation of 0.5. The graphs would

Approximate Binomial Probabilities Using the Normal Distribution -According to government data, the probability than an adult never had the flu is 11%. You randomly select 53 adults and ask if he or she ever had the flu. Decide whether you can use the normal distribution to approximate the binomial distribution, If so, find the mean and standard deviation, If not, explain why.

Provide an appropriate response -In a uniform probability distribution, any random variable is just as likely as any other random variable to occur, provided the random variables belong to the distribution.

Find z-scores for a Given Area -For a standard normal curve, find the z-score that separates the bottom $90 \%$ from the top $10 \%$ .