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Chapter 7: Problem 92

Provide graphs and use the calculator. Men have an average weight of 172 pounds with a standard deviation of 29 pounds. a. Find the probability that 20 randomly selected men will have a sum weight greater than 3600 lbs. b. If 20 men have a sum weight greater than 3500 lbs, then their total weight exceeds the safety limits for water taxis. Based on (a), is this a safety concern? Explain.

Short Answer

Expert verified
The probability of exceeding the weight is 0.1075, indicating a safety concern.

Step by step solution

01

Calculate the Mean and Standard Deviation for the Sample Sum

First, we need to determine the mean and standard deviation of the sum of the weights for 20 men. Each man's weight, \(X_i\), is distributed with a mean \(\mu = 172\) and a standard deviation \(\sigma = 29\). If \(S\) is the sum of the weights of 20 men, then the mean of \(S\) is given by \(\mu_S = n \times \mu\) where \(n = 20\). Thus, \(\mu_S = 20 \times 172 = 3440\). Next, calculate the standard deviation of the sum, \(\sigma_S = \sqrt{n} \times \sigma = \sqrt{20} \times 29 = 129.29\).
02

Find the Z-Score for the Sum Weight Greater than 3600 lbs

Use the normal distribution to find the probability that the sum of weights is greater than 3600 lbs. We need to calculate the Z-score for 3600 lbs using the mean and standard deviation from Step 1:\[ Z = \frac{3600 - \mu_S}{\sigma_S} = \frac{3600 - 3440}{129.29} = 1.24 \]This Z-score will help us find the probability using the standard normal distribution.
03

Calculate the Probability

Using a Z-table, calculator, or software, find the probability that Z > 1.24. From the Z-table, the cumulative probability for Z = 1.24 is approximately 0.8925. Therefore, the probability that a randomly selected sample of 20 men would have a total weight greater than 3600 pounds is:\[ P(Z > 1.24) = 1 - 0.8925 = 0.1075 \]
04

Analyze the Safety Concern

Since the probability of the sum of weights being greater than 3500 lbs (which is a safety concern) is based on the calculation of exceeding 3600 lbs, which has a probability of 0.1075 (greater than typical significance levels, such as 0.05), there is a significant chance of exceeding the safety limit. This indicates a potential safety concern.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal distribution
Understanding the concept of normal distribution is essential when dealing with datasets that exhibit a bell-shaped curve. In statistics, a normal distribution is a common pattern for the spread of a dataset, where most outcomes cluster around a central peak, tapering symmetrically as they move away from the center.
When a dataset follows a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This is known as the empirical rule or the 68-95-99.7 rule.
  • A common example observed in human populations is attributes like height or weight, where the majority of the data is centered around an average value.
  • The normal distribution is pivotal in calculating probabilities and making inferences about a population from sample data.
  • In our exercise, the weights are assumed to follow a normal distribution, enabling the use of statistical tools like Z-scores for probability estimation.
Z-score calculation
Z-scores are vital statistical tools used to determine how far away an element is from the mean, expressed in terms of standard deviations. They are crucial when trying to understand where a particular score lies within a normal distribution.
To find a Z-score, one can use the formula:\(Z = \frac{X - \mu}{\sigma} \), where \(X\) is the value being assessed, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
  • The Z-score tells you how many standard deviations away an element is from the mean.
  • If a Z-score is 0, it indicates the element is exactly at the mean.
  • Positive Z-scores indicate values above the mean, while negative Z-scores indicate values below the mean.
  • In our exercise, we calculated a Z-score to determine the probability of a group of men having a combined weight greater than a specified amount.
This calculation is the cornerstone for finding probabilities in a normal distribution, as it allows one to use the standard normal distribution table to find probabilities associated with different outcomes.
Standard deviation
Standard deviation is a fundamental concept in statistics that measures the amount of variation or dispersion in a set of values. It indicates how much individual data points differ from the mean.
  • A low standard deviation means that data points tend to be close to the mean.
  • A high standard deviation indicates that data points are spread out over a wider range of values.
  • Mathematically, it is calculated by taking the square root of the variance. The formula for standard deviation for a sample is:\[\sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \]
Where:
  • \(X_i\) is each data point in the dataset.
  • \(\mu\) is the mean of all data points.
  • \(N\) is the number of data points.
In the context of our exercise, knowing the standard deviation of individual weights allows us to calculate the standard deviation of the sum of multiple individuals' weights. This is crucial for determining the range of possible combined weights for a group and calculating related probabilities.
Mean calculation
Calculating the mean, or average, is one of the simplest yet most powerful tools in statistics. It provides a single value that represents the central point of a dataset.
  • The mean is calculated by summing all the values in the dataset and then dividing by the number of values.
  • The formula for finding the mean \(\mu\) is: \(\mu = \frac{\sum X_i}{N}\) where \(X_i\) represents each individual value and \(N\) is the total number of values.
In statistical analysis, especially when dealing with normal distributions, the mean serves as an indicator of the dataset's central tendency.
  • It allows for comparison with other statistical measures like median and mode.
  • For a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.
In the exercise example, we used the mean of individual men's weights to calculate the expected mean of the combined weight for a group, a vital step in further statistical assessments and probability calculations.

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Most popular questions from this chapter

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