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.59b

Chapter 5, Problem 5.R.59b

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.

Verified step by step guidance

Step 1: Identify the key information provided in the problem. The population mean (μ) is \$72,000, the population standard deviation (σ) is \$11,000, the sample size (n) is 45, and we are tasked with finding the probability that the sample mean (x̄) is greater than \$68,000.

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 = \$72,000), 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 = \$11,000 and n = 45. This will give you the standard error, which represents the spread of the sampling distribution.

Step 4: Convert the sample mean ($68,000) to a z-score using the formula z = (x̄ - μ) / SE. Substitute the values: x̄ = $68,000, μ = \$72,000, and the SE calculated in Step 3. The z-score represents how many standard errors the sample mean is below the population mean.

Step 5: Use the z-score obtained in Step 4 to find the corresponding probability from the standard normal distribution table or a statistical software. Since the problem asks for the probability that the sample mean is more than \$68,000, calculate 1 - P(Z ≤ z), where P(Z ≤ z) is the cumulative probability up to 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, with a population standard deviation of $11,000 and a sample size of 45, the SE helps determine how much the sample mean is expected to vary from the true mean.

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 scenario, calculating the Z-score for a sample mean of $68,000 allows us to find the corresponding probability using the standard normal distribution.

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Related Practice

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 (c) greater than 60.

Textbook Question

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

Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (c) more than 11.1 picomoles per liter. Compare your answers with those in Exercise 34.

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 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.

Textbook Question

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

The mean ACT composite score in a recent year is 20.7. A random sample of 36 ACT composite scores is selected. What is the probability that the mean score for the sample is (a) less than 22, (b) greater than 23, and (c) between 20 and 21.5? Assume sigma=5.9.