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.60b

Chapter 5, Problem 5.R.60b

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.

Verified step by step guidance

Step 1: Identify the key information provided in the problem. The population mean (μ) is \$87,000, the population standard deviation (σ) is \$10,500, the sample size (n) is 50, and we are tasked with finding the probability that the sample mean (x̄) is greater than \$85,000.

Step 2: Recognize that the sampling distribution of the sample mean follows a normal distribution because the sample size is sufficiently large (n > 30). The mean of the sampling distribution is equal to the population mean (μ = \$87,000), and the standard error of the mean (SE) is calculated as SE = σ / √n.

Step 3: Calculate the standard error of the mean using the formula SE = σ / √n. Substitute the values: σ = \$10,500 and n = 50. This will give you the standard error.

Step 4: Convert the sample mean ($85,000) to a z-score using the formula z = (x̄ - μ) / SE. Substitute the values: x̄ = $85,000, μ = \$87,000, and the SE calculated in the previous step.

Step 5: Use the z-score obtained in Step 4 to find the corresponding probability from the standard normal distribution table. Since the problem asks for the probability that the sample mean is greater than \$85,000, calculate the area to the right of the z-score. Interpret the result in the context of the problem.

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Key Concepts

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

Central Limit Theorem

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's distribution, provided the sample size is sufficiently large (typically n > 30). This theorem is crucial for calculating probabilities related to sample means, especially when dealing with large samples.

Standard Error

The Standard Error (SE) measures the dispersion of sample means around the population mean. It is calculated as the population standard deviation divided by the square root of the sample size (SE = sigma / √n). In this case, it helps determine how much variability we can expect in the sample mean of physical therapists' salaries.

Z-Score

A Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula Z = (X - μ) / SE, where X is the value of interest, μ is the population mean, and SE is the standard error. In this context, the Z-score will help assess the probability of the sample mean being greater than $85,000 by comparing it to the expected mean salary.

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Related Practice

Textbook Question

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

The mean annual salary for Level 1 actuaries in the United States is about \$72,000. A random sample of 45 Level 1 actuaries is selected. What is the probability that the mean annual salary of the sample is (b) more than $68,000? Assume sigma = $11,000.

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

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.

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