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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 9c
Chapter 6, Problem 9c

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.

Verified step by step guidance
1
Step 1: Identify the population and the sample size. The population is {4, 5, 9}, and the sample size is n = 2. Sampling is done with replacement, meaning each element can be selected more than once.
Step 2: List all possible samples of size 2. Since sampling is with replacement, the possible samples are: (4, 4), (4, 5), (4, 9), (5, 4), (5, 5), (5, 9), (9, 4), (9, 5), (9, 9).
Step 3: For each sample, calculate the median. The median of a sample is the middle value when the sample is ordered. For example, the median of (4, 5) is 4.5, and the median of (5, 9) is 7. Repeat this for all samples.
Step 4: Construct the sampling distribution of the sample median. This involves listing all unique median values and their corresponding probabilities. The probability of each median value is determined by the frequency of occurrence of that median divided by the total number of samples.
Step 5: Calculate the mean of the sampling distribution of the sample median. Use the formula for the mean of a probability distribution: \( \mu = \sum (x_i \cdot P(x_i)) \), where \(x_i\) represents each unique median value and \(P(x_i)\) is its probability. Sum the products of each median value and its probability to find the mean.

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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 median) obtained from a large number of samples drawn from a specific population. It describes how the statistic varies from sample to sample and is crucial for understanding the behavior of sample statistics in inferential statistics.
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Sampling Distribution of Sample Proportion

Sample Median

The sample median is the middle value of a sample when the data points are arranged in ascending order. For an even number of observations, it is the average of the two middle values. The median is a measure of central tendency that is less affected by outliers than the mean, making it useful in skewed distributions.
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Sampling Distribution of Sample Proportion

Mean of the Sampling Distribution

The mean of the sampling distribution of a statistic is the expected value of that statistic across all possible samples. For the sample median, this mean provides insight into the central tendency of the sample medians derived from the population, allowing for predictions about the sample median's behavior in repeated sampling.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

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.)

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

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.


a. For a randomly selected subject, find the probability of a bone density test score greater than -1.37.

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.