Exam 6: Normal Probability Distributions

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 5, 7, and 9. Consider the values of 5, 7, and 9 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 5, 7, and 9. The nine different samples are as follows: (5, 5), (5, 7), (5, 9), (7, 5), (7, 7), (7, 9), (9, 5), (9, 7), and (9, 9). (i)Find the mean of each of the nine samples, then summarize the sampling distribution of the means in the format of a table representing the probability distribution. (ii)Compare the population mean to the mean of the sample means. (iii)Do the sample means target the value of the population mean? In general, do means make good estimators of population means? Why or why not?

Estimate the probability of getting exactly 43 boys in 90 births. Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.

The continuity correction is used to compensate for the fact that a ___________distribution is used to approximate a ____________ distribution.

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

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 no more than 35 defective CDs

Scores on a test have a mean of 70 and $Q _ { 3 }$ is 83 . The scores have a distribution that is approximately normal. Find $P _ { 90 }$ (You will need to first find the standard deviation.)

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

Suppose that you wish to find 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.

SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of 120 (based on data from the College Board ATP). (a)If a single student is randomly selected, find the probability that the sample mean is above 500. (b)If a sample of 35 students are selected randomly, find the probability that the sample mean is above 500. These two problems appear to be very similar. Which problem requires the application of the central limit theorem, and in what way does the solution process differ between the two problems?

State the central limit theorem. Describe the sampling distribution for a population that is uniform and for a population that is normal.

A normal quartile plot is given below for the lifetimes (in hours)of a sample of batteries of a particular brand. Use the plot to assess the normality of the lifetimes of these batteries. Explain your reasoning.

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)

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

Define a density curve and describe the two properties that it must satisfy. Show a density curve for a uniform distribution. Make sure that your graph satisfies both properties.

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?

Find the probability that in 200 tosses of a fair die, we will obtain at most 30 fives. Use the normal distribution to approximate the desired probability.

Define the central limit theorem and its relationship to the sampling distribution of sample means. Define how you can approximate a normal distribution from an original population that is not normally distributed

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.

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. Approximately what percentage of women have red blood cell counts in the Normal range from 4.2 to 5.4 million cells per microliter?

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