Exam 6: Normal Probability Distributions

The normal distribution has a greater percentage of its area close to the mean and much less in the tails. -Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). Find the IQ score separating the top 14% from the others.

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. -

In a recent year, the U.S. Mint in Denver manufactured 270 million quarters. Assume that on each day of production, a sample of 50 quarters is randomly selected and the mean weight is obtained. Given that the population of quarters has a mean weight of 5.67 g, what do you know about the mean of the sample means? What do you know about the shape of the distribution of the sample means?

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of $0 ^ { \circ } \mathrm { C }$ at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below $0 ^ { \circ } \mathrm { C }$ (denoted by negative numbers) and some give readings above $0 ^ { \circ } \mathrm { C }$ (denoted by positive numbers). Assume that the mean reading is $0 ^ { \circ } \mathrm { C }$ and the standard deviation of the readings is $1.00 ^ { \circ } \mathrm { C }$ Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information. -Find P $\mathrm { P } _ { 96 }$ the 96th percentile.

Solve the problem. -Scores on a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4. Estimate the probability that among 75 randomly selected students, at least 20 of them score greater than 78.

Solve the problem. -In a game of roulette, Jorge places 170 bets of $1 each on the number 3. A win pays off with odds 35:1 and on any one spin there is a $1 / 38$ probability that 3 will be the winning number. Among the 170 bets, what is the Minimum number of wins needed for Jorge to make a profit? Estimate the probability that Jorge will make a Profit.

Find the indicated value. - $z _ { 0 } .005$

The normal distribution has a greater percentage of its area close to the mean and much less in the tails. -Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). Find the probability that a randomly selected adult has an IQ between 90 and 120 (somewhere in the range of normal to bright normal).

If z is a standard normal variable, find the probability. -P(z > 0.59)

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. -Shaded area is 0.9599.

Scores on a test have a mean of 70 and Q3 is 83. The scores have a distribution that is approximately normal. Find P90. (You will need to first find the standard deviation.)

A coin is tossed 20 times. A person who claims to have extrasensory perception is asked to predict the outcome of each flip in advance. She predicts correctly on 14 tosses. What is the probability of being correct 14 or more Times by guessing? Does this probability seem to verify her claim? Use the normal distribution to approximate The desired probability.

Assume that X has a normal distribution, and find the indicated probability. -The mean is $\mu = 60.0$ .0 and the standard deviation is $\sigma = 4.0$ Find the probability that X is less than 53.0.

The weights of college football players are normally distributed with a mean of 200 pounds and a standard deviation of 50 pounds. If a college football player is randomly selected, find the probability that he weighs Between 170 and 220 pounds.

Solve the problem. -Personal phone calls received in the last three days by a new employee were 3, 6, and 5. Assume that samples of size 2 are randomly selected with replacement from this population of three values. Identify the probability of each sample, and describe the sampling distribution of the sample means.

If z is a standard normal variable, find the probability: $P ( - 0.73 < z < 2.27 )$

Lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. (a) Find the probability of a pregnancy lasting more than 250 days. (b) Find the probability of a pregnancy lasting more than 280 days. Draw the diagram for each and discuss the part of the solution that would be different for finding the requested probabilities.

Solve the problem. -A population consists of these values: 3, 5, 8, 14, 17. Find the mean, mu , and standard deviation, $\sigma$ , of the population. List all samples of size n = 2 that are obtained without replacement. Find the population of all values of $\overline { \mathrm { X } }$ by finding the mean of each sample that you listed. Find the mean $\mu _ { x } ^ { - }$ and standard deviation $\sigma _ { \mathrm { X } } ^ { - }$ for the population of sample means by using the formulas: $\mu _ { \mathrm { x } } ^ { - } = \mu \text { and } \sigma _ { \mathrm { x } } ^ { - } = \frac { \sigma } { \sqrt { \mathrm { n } } } \sqrt { \frac { \mathrm { N } - \mathrm { n } } { \mathrm { N } - 1 } }$

Solve the problem. -A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is greater than 78.

The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. The probability of exactly 44 green marbles