Ergonomics. Exercises 9–16 involve applications to ergonomics, as...

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Introduction to Confidence Intervals

2:10 minutes

Problem 6.4.10a

Textbook Question

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

Designing Manholes According to the website www.torchmate.com, “manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter.” Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder widths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey).

a. What percentage of men will fit into the manhole?

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the percentage of men who can fit into a manhole with a diameter of 22 inches. Since the opening is circular, the relevant dimension for fitting is the shoulder width of men, which is normally distributed with a mean (μ) of 18.2 inches and a standard deviation (σ) of 1.0 inch.

Step 2: Define the threshold for fitting. A man can fit into the manhole if his shoulder width is less than or equal to the diameter of the manhole, which is 22 inches. Therefore, we need to calculate the probability that a randomly selected man has a shoulder width ≤ 22 inches.

Step 3: Standardize the value of 22 inches using the z-score formula. The z-score formula is:

z = \frac{x - μ}{σ}

, where x is the value of interest (22 inches), μ is the mean (18.2 inches), and σ is the standard deviation (1.0 inch). Substitute these values into the formula to compute the z-score.

Step 4: Use the z-score to find the cumulative probability. Once the z-score is calculated, use a standard normal distribution table or statistical software to find the cumulative probability corresponding to that z-score. This cumulative probability represents the proportion of men with shoulder widths ≤ 22 inches.

Step 5: Interpret the result. The cumulative probability obtained in Step 4 is the percentage of men who can fit into the manhole. Multiply the probability by 100 to express it as a percentage.

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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, men's shoulder widths are normally distributed, which allows us to use the mean and standard deviation to determine the percentage of men fitting through the manhole.

Z-Score

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In this scenario, calculating the Z-score for the manhole diameter will help determine how many men have shoulder widths that fall within the allowable range.

Percentile

A percentile is a measure used in statistics indicating the value below which a given percentage of observations fall. In this case, once the Z-score is calculated, we can use the standard normal distribution table to find the corresponding percentile, which will tell us the percentage of men whose shoulder widths are less than or equal to the manhole diameter.

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Key Concepts

Normal Distribution

Z-Score

Percentile

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