Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 11a

Chapter 6, Problem 11a

In Exercises 11–14, use the population of {2, 3, 5, 9} of the lengths of hospital stay (days) of mothers who gave birth, found from Data Set 6 “Births” in Appendix B. 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, and 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.)

Verified step by step guidance

Step 1: Identify all possible samples of size n = 2 from the population {2, 3, 5, 9}. Since sampling is done with replacement, each sample can include repeated values. The total number of possible samples is 4 × 4 = 16. List all 16 samples explicitly, such as (2, 2), (2, 3), (2, 5), ..., (9, 9).

Step 2: For each sample, calculate the sample mean. The sample mean is given by the formula:

x = \frac{(x + y)}{2}

, where x and y are the two values in the sample. Compute the mean for all 16 samples.

Step 3: Construct a frequency table for the sampling distribution of the sample mean. Group together identical sample means and count their frequencies. For example, if the sample mean 2.5 appears 3 times, record it as a frequency of 3.

Step 4: Organize the table to display the unique sample means in one column and their corresponding frequencies in another column. This table represents the sampling distribution of the sample mean.

Step 5: Verify that the sum of the frequencies in the table equals the total number of samples (16). This ensures that all possible samples and their means have been accounted for in the distribution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sampling Distribution

A sampling distribution is the probability distribution of a statistic (like the sample mean) obtained from a large number of samples drawn from a specific population. It illustrates how the sample mean varies from sample to sample, providing insights into the reliability and variability of the estimate. Understanding this concept is crucial for analyzing how well a sample represents the population.

Sample Mean

The sample mean is the average value of a set of observations in a sample. It is calculated by summing all the sample values and dividing by the number of observations. The sample mean serves as an estimator for the population mean, and its distribution is central to inferential statistics, particularly in hypothesis testing and confidence intervals.

Combining Values in a Distribution Table

When constructing a distribution table for sample means, it is important to combine identical values to simplify the representation. This involves counting how many times each unique sample mean occurs and presenting it alongside its frequency. This process helps in visualizing the distribution and understanding the likelihood of different sample means occurring, which is essential for statistical analysis.

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Combinations

Related Practice

Textbook Question

Bone Density Test A bone mineral density test is used to identify a bone disease. The result of a bone density test is commonly measured as a z score, and the population of z scores is normally distributed with a mean of 0 and a standard deviation of 1.

c. For a randomly selected subject, find the probability of a bone density test score between -0.67 and 1.29.

Textbook Question

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

c. Find the mean of the sampling distribution of the sample median.

Textbook Question

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).

d. If a vending machine is designed to accept quarters with weights above the 10th percentile P10 find the weight separating acceptable quarters from those that are not acceptable.

Textbook Question

Quarters Assume that weights of quarters minted after 1964 are normally distributed with a mean of 5.670 g and a standard deviation of 0.062 g (based on U.S. Mint specifications).

a. Find the probability that a randomly selected quarter weighs between 5.600 g and 5.700 g..

Textbook Question

Normal Distribution Using a larger data set than the one given for the preceding exercises, assume that cell phone radiation amounts are normally distributed with a mean of 1.17 W/kg and a standard deviation of 0.29 W/kg.

b. Find the value of Q3, the cell phone radiation amount that is the third quartile.

Textbook Question

Correcting for a Finite Population In a study of babies born with very low birth weights, 275 children were given IQ tests at age 8, and their scores approximated a normal distribution with μ = 95.5 and σ = 16.0 (based on data from “Neurobehavioral Outcomes of School-age Children Born Extremely Low Birth Weight or Very Preterm,” by Anderson et al., Journal of the American Medical Association, Vol. 289, No. 24). Fifty of those children are to be randomly selected without replacement for a follow-up study.

b. Find the probability that the mean IQ score of the follow-up sample is between 95 and 105.