Exam 6: Normal Probability Distributions

A baseball player has a batting average of 0.346, so the probability of a hit is 0.346. Assume that his hitting attempts are independent of each other. Assume that the batter gets up to bat 4 times in each game. Estimate the probability that in 50 consecutive games, there are at least 45 games in which the batter gets at least one hit. (Hint: first find the probability that in one Game the batter gets at least one hit)

Scores on a test are normally distributed with a mean of 63.2 and a standard deviation of 11.7 . Find $P _ { 81 }$ which separates the bottom 81 % from the top 19% .

In a population of 210 women, the heights of the women are normally distributed with a mean of 64.4 inches and a standard deviation of 2.9 inches. If 36 women are selected at random, find the mean $\mu _ { \bar { x } }$ and standard deviation $\sigma _ { \bar { x } }$ of the population of sample means. Assume that the sampling is done without replacement and use a finite population correction factor.

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

Suppose that you wish to find $P ( - 2 < x < 2 )$ for a continuous uniform distribution having a minimum of -3 and a maximum of 3 . If you incorrectly assume that the distribution is normal instead of uniform, will your answer be too big, too small, or will you still obtain the correct answer? Explain your thinking.

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 2, 3, and 8. Consider the values of 2, 3, and 8 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and 8. The nine different samples are as follows: (2, 2), (2, 3), (2, 8), (3, 2), (3, 3), (3, 8), (8, 2), (8, 3), and (8, 8). (i)Find the range of each of the nine samples, then summarize the sampling distribution of the ranges in the format of a table representing the probability distribution. (ii)Compare the population range to the mean of the sample ranges. (iii)Do the sample ranges target the value of the population range? In general, do ranges make good estimators of population ranges? Why or why not?

For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg. If 23 women aged 18-24 are randomly selected, find the probability that their mean systolic blood pressure is between 119 and 122 mm Hg.

If a sample size is < _____, the sample size must come from a population having a normal distribution in order to follow normal distribution calculations.

Heights of adult females are normally distributed. Samples of height of adult females, each of size n=3 , are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as a normal distribution because the sample size is so small? Explain.

Explain how a nonstandard normal distribution differs from the standard normal distribution. Describe the process for finding probabilities for nonstandard normal distributions.

The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy last at least 300 days?

After constructing a new manufacturing machine, five prototype integrated circuit chips are produced and it is found that two are defective and three are acceptable. Assume that two of the chips are randomly selected with replacement from this population. After identifying the 25 possible samples, find the proportion of defects in each of them, using a table to describe the sampling distribution of the proportions of the defects.

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. Two percent of hair dryers produced in a certain plant are defective. Estimate the probability that of 10,000 randomly selected hair dryers, at least 219 Are defective.

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?

Assume that the red blood cell counts of women are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells per microliter. Find the 80^th percentile for the red blood cell counts of women.

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 the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability that the given range of pounds lost is between 8 pounds and 11 pounds.

If a histogram of a sample of men's ages is skewed, what do you expect to see in the normal quantile plot?

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.

A normal quartile plot is given below for the weekly incomes (in dollars)of a sample of engineers in one town. Describe what each x value represents and what each y value represents. Use the plot to assess the normality of the incomes of engineers in this town. Explain your reasoning.