Ch. 5 - Normal Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 5 - Normal Probability Distributions

Problem 5.T.6

Chapter 5, Problem 5.T.6

Use technology to find the standard deviation of the set of 36 sample means. How does it compare with the standard deviation of the ages found in Exercise 5? Does this agree with the result predicted by the Central Limit Theorem?

Verified step by step guidance

Step 1: Recall the formula for the standard deviation of the sampling distribution of the sample mean (also known as the standard error of the mean). It is given by:

σ

_mean = σ / n, where

σ

is the population standard deviation and

n

is the sample size.

Step 2: Identify the values needed for the formula. From the problem, the sample size

n

is 36. You will also need the population standard deviation

σ

, which should be provided in Exercise 5 or the dataset.

Step 3: Use technology (such as a calculator, spreadsheet software, or statistical software) to compute the standard deviation of the sample means. Plug the values of

σ

and

n

into the formula:

σ

_mean = σ / n.

Step 4: Compare the computed standard deviation of the sample means to the standard deviation of the ages found in Exercise 5. Note that the standard deviation of the sample means should be smaller than the population standard deviation, as predicted by the Central Limit Theorem.

Step 5: Reflect on the Central Limit Theorem, which states that the standard deviation of the sampling distribution of the sample mean decreases as the sample size increases. Verify that the result aligns with this prediction, as the sample size of 36 is relatively large.

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Key Concepts

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

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. It is crucial for understanding the variability of sample means in relation to the population from which they are drawn.

Central Limit Theorem (CLT)

The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size increases, regardless of the original distribution of the population. This theorem is fundamental in statistics because it allows for the use of normal probability techniques to make inferences about population parameters based on sample statistics, especially when dealing with large samples.

Sample Means

Sample means are the averages calculated from subsets of a population. When multiple samples are taken, each will yield a mean, and analyzing these sample means can provide insights into the overall population mean. The standard deviation of these sample means, known as the standard error, is essential for understanding how much the sample means are expected to vary from the true population mean.

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Related Practice

Textbook Question

In Exercises 5 and 6, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities.

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (c) at least 10. Identify any unusual events. Explain.

Textbook Question

Assume the machine shifts and the distribution of the amount of the compound added now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added.

a. What is the probability that you select a vial that is within the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.)

Textbook Question

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (a) exactly 7. Identify any unusual events. Explain.

Textbook Question

A survey of U.S. undergraduates found that 37% of those attending in-state colleges would prefer to take a job in a different state after graduation. You randomly select 18 U.S. undergraduates attending in-state colleges. Find the probability that the number who would prefer to take a job in a different state after graduation is (b) less than 5. Identify any unusual events. Explain.

Textbook Question

Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added.

a. What is the probability that you select a sample of five vials that has a mean that is within the acceptable range? (See figure.)

Textbook Question

In Exercises 2–4, the random variable x is normally distributed with mean mu= 18 and standard deviation sigma 7.6

Find the value of x that has 88.3% of the distribution’s area to its left.