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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 6.3.7c
Chapter 6, Problem 6.3.7c

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.


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

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Identify the population {4, 5, 9} and note that the sample size n = 2, with sampling done with replacement.
List all possible samples of size 2 from the population. 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).
For each sample, calculate the sample variance using the formula: \( s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2 \), where \( \bar{x} \) is the sample mean and \( n \) is the sample size.
Construct the sampling distribution of the sample variance by listing all the calculated variances and their corresponding probabilities. Since sampling is with replacement, each sample has an equal probability of \( \frac{1}{9} \).
Find the mean of the sampling distribution of the sample variance by using the formula: \( \mu_{s^2} = \sum (s^2 \cdot P(s^2)) \), where \( s^2 \) represents each sample variance and \( P(s^2) \) is its probability.

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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 or variance) 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 estimators in statistics.
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Sampling Distribution of Sample Proportion

Sample Variance

Sample variance is a measure of how much the values in a sample differ from the sample mean. It is calculated by taking the average of the squared differences between each data point and the sample mean. Understanding sample variance is essential for assessing the variability within a sample and for making inferences about the population.
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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 variance, this mean provides insight into the average variability one can expect when taking samples from a population, and it is a key component in inferential statistics.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).


35 35 20 50 95 75 45 50 30 35 30 30


Tower of Terror Wait Times


a. Find Q1, Q2 and Q3.

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Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).


35 35 20 50 95 75 45 50 30 35 30 30


g. What level of measurement (nominal, ordinal, interval, ratio) describes this data set?

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Textbook Question

Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores.


c. If the grades are curved so that grades of B are given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B.

Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.


Safe Loading of Elevators The elevator in the car rental building at San Francisco International Airport has a placard stating that the maximum capacity is “4000 lb—27 passengers.” Because 4000/27=148, this converts to a mean passenger weight of 148 lb when the elevator is full. We will assume a worst-case scenario in which the elevator is filled with 27 adult males. Based on Data Set 1 “Body Data” in Appendix B, assume that adult males have weights that are normally distributed with a mean of 189 lb and a standard deviation of 39 lb.


c. What do you conclude about the safety of this elevator?

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 Proportion


c. Find the mean of the sampling distribution of the sample proportion of odd numbers.

Textbook Question

Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population.


c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of peas with yellow pods? Does the mean of the sampling distribution of proportions always equal the population proportion?