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.R.53

Chapter 5, Problem 5.R.53

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

The population densities in people per square mile in the 50 U.S. states have a mean of 199.6 and a standard deviation of 265.4. Random samples of size 35 are drawn from this population, and the mean of each sample is determined.

Verified step by step guidance

Step 1: Recall the formulas for the mean and standard deviation of the sampling distribution of sample means. The mean of the sampling distribution (μₓ̄) is equal to the population mean (μ), and the standard deviation of the sampling distribution (σₓ̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n).

Step 2: Substitute the given values into the formulas. The population mean (μ) is 199.6, the population standard deviation (σ) is 265.4, and the sample size (n) is 35. Use the formula for the standard deviation of the sampling distribution: σₓ̄ = σ / √n.

Step 3: Simplify the expression for the standard deviation of the sampling distribution. Calculate the square root of the sample size (√n) and divide the population standard deviation (σ) by this value.

Step 4: The mean of the sampling distribution (μₓ̄) is the same as the population mean, so μₓ̄ = 199.6. The standard deviation of the sampling distribution (σₓ̄) is the value obtained in Step 3.

Step 5: To sketch the graph of the sampling distribution, draw a normal distribution curve centered at the mean (199.6) with the standard deviation (σₓ̄) calculated in Step 3. Label the x-axis with values around the mean, spaced by increments of the standard deviation.

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Key Concepts

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

Sampling Distribution of Sample Means

The sampling distribution of sample means refers to the distribution of the means of all possible samples of a specific size drawn from a population. It is crucial for understanding how sample means behave, particularly in relation to the population mean. According to the Central Limit Theorem, as the sample size increases, the sampling distribution approaches a normal distribution, regardless of the population's distribution, provided the sample size is sufficiently large.

Mean of the Sampling Distribution

The mean of the sampling distribution of sample means, also known as the expected value, is equal to the population mean. In this case, it is 199.6 people per square mile. This concept is essential for predicting the average outcome of sample means and is foundational for inferential statistics, allowing statisticians to make generalizations about the population based on sample data.

Standard Deviation of the Sampling Distribution (Standard Error)

The standard deviation of the sampling distribution, often referred to as the standard error, measures the variability of sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. In this scenario, the standard error helps assess how much the sample means are expected to fluctuate, providing insight into the reliability of the sample estimates.

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Related Practice

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean MCAT total score in a recent year is 500.9. A random sample of 32 MCAT total scores is selected. What is the probability that the mean score for the sample is (b) more than 502? Assume sigma=10.6.

Textbook Question

In Exercises 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

Textbook Question

Determine whether any of the events in Exercise 33 are unusual. Explain your reasoning.

Textbook Question

In Exercises 69 and 70, 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. adults found that 72% used a mobile device to manage their bank account at least once in the previous month. You randomly select 70 U.S. adults and ask whether they used a mobile device to manage their bank account at least once in the previous month. Find the probability that the number who have done so is (a) at most 40.

Textbook Question

In Exercises 61 and 62, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

A survey of U.S. adults ages 33 to 40 earning more than \$150,000 per year found that 94% are content with how their lives have turned out so far. You randomly select 20 U.S. adults ages 33 to 40 earning more than \$150,000 and ask if they are content with their lives so far.

Textbook Question

In Exercises 55–60, find the indicated probabilities and interpret the results.

The mean annual salary for physical therapists in the United States is about \$87,000. A random sample of 50 physical therapists is selected. What is the probability that the mean annual salary of the sample is (b) more than $85,000? Assume sigma = $10,500.