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Chapter 6: Q2 (page 240)

Normal Distribution A normal distribution is informally described as a probability distribution that is “bell-shaped” when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.

Short Answer

Expert verified

The rough sketch of the normal curve is shown below.

Step by step solution

01

Given information

A normal distribution is bell-shaped.

02

Define a normal distribution

A normal distribution shows a bell shape where the curve rises from left to right, reaching the height at the mean value and then decreasing symmetrically.

03

Step 3:Sketch the normal curve

The sketch of a normal curve is shown below.

The graph is sketched on two axes; the values of a random variable and the corresponding values of the random variables of normal distribution and the probabilities.

The graph is sketched below.

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Most popular questions from this chapter

In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

Requirements A researcher collects a simple random sample of grade-point averages of statistics students, and she calculates the mean of this sample. Under what conditions can that sample mean be treated as a value from a population having a normal distribution?

Birth Weights Based on Data Set 4 “Births” in Appendix B, birth weights are normally distributed with a mean of 3152.0 g and a standard deviation of 693.4 g.

a. For the bell-shaped graph, what is the area under the curve?

b. What is the value of the median?

c. What is the value of the mode?

d. What is the value of the variance?

In Exercises 11–14, use the population of {34, 36, 41, 51} of the amounts of caffeine (mg/12oz) in Coca-Cola Zero, Diet Pepsi, Dr Pepper, and Mellow Yello Zero.

Assume that  random samples of size n = 2 are selected with replacement.

Sampling Distribution of the Sample Mean

a. After identifying the 16 different possible samples, find the mean of each sample, then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same. (Hint: See Table 6-3 in Example 2 on page 258.)

b. Compare the mean of the population {34, 36, 41, 51} to the mean of the sampling distribution of the sample mean.

c. Do the sample means target the value of the population mean? In general, do sample means make good estimators of population means? Why or why not?

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

Sampling Distribution of the Sample Median

a. Find the value of the population median.

b. Table 6-2 describes the sampling distribution of the sample mean. Construct a similar table representing the sampling distribution of the sample median. Then combine values of the median that are the same, as in Table 6-3. (Hint: See Example 2 on page 258 for Tables 6-2 and 6-3, which describe the sampling distribution of the sample mean.)

c. Find the mean of the sampling distribution of the sample median. d. Based on the preceding results, is the sample median an unbiased estimator of the population median? Why or why not?
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