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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 6.3.18c
Chapter 6, Problem 6.3.18c

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?

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Step 1: Begin by understanding the problem. The population consists of 4 peas with yellow pods and 1 pea with a green pod. This means the population proportion of peas with yellow pods is p = 4/5 = 0.8. The question asks about the mean of the sampling distribution of proportions and whether it equals the population proportion.
Step 2: Recall the formula for the mean of the sampling distribution of proportions. The mean of the sampling distribution of proportions (denoted as μ_p̂) is equal to the population proportion (p). Mathematically, μ_p̂ = p.
Step 3: Since the sampling is done with replacement, the probability of selecting a yellow pod remains constant at 0.8 for each draw. This ensures that the sampling distribution of proportions is unbiased, and its mean will equal the population proportion.
Step 4: Address the second part of the question. The mean of the sampling distribution of proportions always equals the population proportion, provided the sampling is random and unbiased. This is a fundamental property of the sampling distribution of proportions.
Step 5: Conclude that in this specific case, the mean of the sampling distribution of proportions is indeed equal to the population proportion of peas with yellow pods, which is 0.8. This result holds true for any random sampling process with replacement.

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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 obtained from a larger population. It describes how the sample statistic (like the sample mean or proportion) varies from sample to sample. In this context, it refers to the distribution of the proportions of yellow pods obtained from repeated random samples of the pea population.
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Sampling Distribution of Sample Proportion

Mean of the Sampling Distribution

The mean of the sampling distribution, also known as the expected value, is the average of all possible sample means or proportions. According to the Central Limit Theorem, this mean will equal the population mean or proportion when the sample size is sufficiently large, regardless of the population's distribution. In this case, it questions whether the mean of the sampling distribution of proportions matches the population proportion of yellow pods.
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Law of Large Numbers

The Law of Large Numbers states that as the size of a sample increases, the sample mean will get closer to the population mean. This principle underpins the idea that the mean of the sampling distribution of proportions will converge to the true population proportion as more samples are taken. It emphasizes the reliability of sample statistics in estimating population parameters when the sample size is large.
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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


c. Find the standard deviation s.

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Textbook Question

In Exercises 25–28, use these parameters (based on Data Set 1 “Body Data” in Appendix B):

Men’s heights are normally distributed with mean 68.6 in. and standard deviation 2.8 in.

Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

If the Navy changes the height requirements so that all women are eligible except the shortest 3% and the tallest 3%, what are the new height requirements for women?

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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.

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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.


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

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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.

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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.


b. Find the mean of the sampling distribution.

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