6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:27 minutesProblem 36a
Textbook Question
Water Taxi Safety When a water taxi sank in Baltimore’s Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a “worst-case” scenario in which all of the passengers are adult men. Assume that weights of men are normally distributed with a mean of 188.6 lb and a standard deviation of 38.9 lb (based on Data Set 1 “Body Data” in Appendix B).
a. If one man is randomly selected, find the probability that he weighs less than 174 lb (the new value suggested by the National Transportation and Safety Board).
Verified step by step guidance1
Step 1: Identify the key components of the problem. The weights of men are normally distributed with a mean (μ) of 188.6 lb and a standard deviation (σ) of 38.9 lb. We need to find the probability that a randomly selected man weighs less than 174 lb.
Step 2: Standardize the value of 174 lb using the z-score formula. The z-score formula is given by: , where x is the value of interest, μ is the mean, and σ is the standard deviation.
Step 3: Substitute the values into the z-score formula. Here, x = 174, μ = 188.6, and σ = 38.9. Calculate the z-score using these values.
Step 4: Use the z-score obtained in Step 3 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. The cumulative probability represents the likelihood that a randomly selected man weighs less than 174 lb. This probability is the solution to the problem.
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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, defined by its mean and standard deviation. In this context, the weights of men are assumed to follow a normal distribution, which allows us to calculate probabilities related to their weights.
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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 a weight of 174 lb will help determine how many standard deviations this weight is from the mean, which is essential for finding the corresponding probability.
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Probability
Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this case, we are interested in the probability that a randomly selected man weighs less than 174 lb. This involves using the normal distribution and Z-scores to find the area under the curve to the left of the calculated Z-score, which represents the desired probability.
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