Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.CRE.3d

Chapter 6, Problem 6.CRE.3d

Foot Lengths of Women Assume that foot lengths of adult females are normally distributed with a mean of 246.3 mm and a standard deviation of 12.4 mm (based on Data Set 3 “ANSUR II 2012” in Appendix B).

d. Find the probability that 16 adult females have foot lengths with a mean greater than 250 mm.

Verified step by step guidance

Step 1: Identify the given parameters. The population mean (μ) is 246.3 mm, the population standard deviation (σ) is 12.4 mm, the sample size (n) is 16, and we are tasked with finding the probability that the sample mean is greater than 250 mm.

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.

Step 3: Standardize the sample mean to a z-score. Use the formula z = (X̄ - μ) / SE, where X̄ is the sample mean (250 mm), μ is the population mean, and SE is the standard error calculated in Step 2.

Step 4: Use the z-score to find the corresponding probability. Look up the z-score in a standard normal distribution table or use statistical software to find the cumulative probability up to the z-score.

Step 5: Subtract the cumulative probability from 1 to find the probability that the sample mean is greater than 250 mm. This is because we are interested in the area to the right of the z-score.

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Key Concepts

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

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In this context, the foot lengths of adult females follow a normal distribution characterized by a specific mean and standard deviation, which allows for the calculation of probabilities related to the data.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will be normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large (typically n > 30). In this case, since we are dealing with a sample of 16 adult females, we can still apply the theorem to approximate the distribution of the sample mean, but we must consider the standard error of the mean.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In this problem, calculating the Z-score for the sample mean of foot lengths will help determine the probability that the mean foot length of 16 adult females exceeds 250 mm.

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Related Practice

Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

b. Construct a boxplot.

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Textbook Question

Normal Distribution Using a larger data set than the one given for the preceding exercises, assume that cell phone radiation amounts are normally distributed with a mean of 1.17 W/kg and a standard deviation of 0.29 W/kg.

a. Find the probability that a randomly selected cell phone has a radiation amount that exceeds the U.S. standard of 1.6 W/kg or less.

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Textbook Question

Foot Lengths of Women Assume that foot lengths of adult females are normally distributed with a mean of 246.3 mm and a standard deviation of 12.4 mm (based on Data Set 3 “ANSUR II 2012” in Appendix B).

c. Find P95.

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Textbook Question

Blue Eyes Assume that 35% of us have blue eyes (based on a study by Dr. P. Soria at Indiana University).

b. Find the value of P(B_bar).

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Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

a. Find the mean xbar.

106

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Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

b. Find the median.

132

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