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 5.4.15

Chapter 5, Problem 5.4.15

Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.

For a random sample of n=64, find the probability of a sample mean being less than 24.3 when Mu=24 and sigma=1.25.

Verified step by step guidance

Step 1: Identify the given values in the problem. The population mean (μ) is 24, the population standard deviation (σ) is 1.25, the sample size (n) is 64, and we are tasked with finding the probability that the sample mean (x̄) is less than 24.3.

Step 2: Calculate the standard error of the mean (SE). The formula for the standard error is SE = σ / √n. Substitute the given values for σ and n into the formula to compute SE.

Step 3: Convert the sample mean (x̄ = 24.3) to a z-score using the formula z = (x̄ - μ) / SE. Substitute the values for x̄, μ, and SE into the formula to calculate the z-score.

Step 4: Use the z-score to find the corresponding probability. Refer to the standard normal distribution table (or use statistical software) to find the cumulative probability associated with the calculated z-score.

Step 5: Interpret the result. Compare the probability to the threshold for unusual events (commonly 0.05 or 5%). If the probability is less than this threshold, the sample mean would be considered unusual; otherwise, it would not be considered unusual.

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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 sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution. This theorem is crucial for understanding how to calculate probabilities related to sample means, especially when the sample size is large, such as n=64 in this case.

Standard Error

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 scenario, the SE will help determine how likely it is for the sample mean to be less than 24.3 given the population parameters.

Z-Score

A Z-score indicates how many standard deviations an element is from the mean. It is calculated using the formula Z = (X - Mu) / SE, where X is the sample mean. By calculating the Z-score for the sample mean of 24.3, we can assess whether this value is unusual compared to the expected distribution of sample means.

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Related Practice

Textbook Question

"Getting Physical The figure shows the results of a survey of U.S. adults ages 18 to 29 who were asked whether they participated in a sport. In the survey, 48% of the men and 23% of the women said they participate in sports. The most common sports are shown below. Use this information in Exercises 29 and 30.

You randomly select 250 U.S. men ages 18 to 29 and ask them whether they participate in at least one sport. You find that 80% say no. How likely is this result? Do you think this sample is a good one? Explain your reasoning."

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Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

P1.5

151

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Textbook Question

Given the mean of a normal distribution, how can you find the median?

155

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Textbook Question

Which Is More Likely? Assume that the fertility rates in Exercise 32 are normally distributed. Are you more likely to randomly select a state with a fertility rate of less than 65 or to randomly select a sample of 15 states in which the mean of the state fertility rates is less than 65? Explain.

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Textbook Question

Finding Probabilities In Exercises 15–18, the population mean and standard deviation are given. Find the indicated probability and determine whether the given sample mean would be considered unusual.

For a random sample of n=45, find the probability of a sample mean being greater than 551 when mu=550 and sigma=3.7.

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Textbook Question

Finding a z-Score In Exercises 1–16, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

0.6736

257

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