Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.q.9

Chapter 6, Problem 6.q.9

Seat Designs. In Exercises 7–9, assume that when seated, adult males have back-to-knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats.

Find the probability that nine males have back-to-knee lengths with a mean greater than 23.0 in.

Verified step by step guidance

Step 1: Identify the given information. The population mean (μ) is 23.5 inches, the population standard deviation (σ) is 1.1 inches, and the sample size (n) is 9. We are tasked with finding the probability that the sample mean (x̄) is greater than 23.0 inches.

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: Standardize the sample mean using the z-score formula. The z-score formula for a sample mean is z = (x̄ - μ) / SE. Substitute the values for x̄ (23.0), μ (23.5), and the calculated SE into the formula to find the z-score.

Step 4: Use the standard normal distribution table (or a statistical software) to find the cumulative probability corresponding to the calculated z-score. This gives the probability that the sample mean is less than 23.0 inches.

Step 5: Subtract the cumulative probability from 1 to find the probability that the sample mean is greater than 23.0 inches. This is the final probability we are looking for.

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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, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean and the standard deviation. In this context, the back-to-knee lengths of adult males are normally distributed, which allows for the application of statistical methods to calculate probabilities.

Central Limit Theorem

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution. This theorem is crucial when dealing with sample means, as it allows us to use normal distribution properties to find probabilities related to sample means, even when the original data may not be perfectly normal.

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, Z-scores can be used to determine the probability of the sample mean of back-to-knee lengths being greater than a specified value, allowing for the assessment of how unusual or typical that sample mean is.

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Textbook Question

Bone Density Test A bone mineral density test is used to identify a bone disease. The result of a bone density test is commonly measured as a z score, and the population of z scores is normally distributed with a mean of 0 and a standard deviation of 1.

c. For a randomly selected subject, find the probability of a bone density test score between -0.67 and 1.29.

Textbook Question

a. For a randomly selected subject, find the probability of a bone density test score greater than -1.37.

Textbook Question

Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.

Bone Density For a randomly selected subject, find the probability of a bone density score between and 2.00.

Textbook Question

Bone Density Test. In Exercises 1–4, assume that scores on a bone mineral density test are normally distributed with a mean of 0 and a standard deviation of 1.

Bone Density Find the bone density score that is the 90th percentile, which is the score separating the lowest 90% from the top 10%.

Textbook Question

Seat Designs. In Exercises 7–9, assume that when seated, adult males have back-to-knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats.

Find the probability that a male has a back-to-knee length greater than 25.0 in.

Textbook Question

Ergonomics. Exercises 9–16 involve applications to ergonomics, as described in the Chapter Problem.

Water Taxi Safety Passengers died when a water taxi sank in Baltimore’s Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 “Body Data” in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb.

d. Is the new capacity of 20 passengers safe?