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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 6.1.50
Chapter 6, Problem 6.1.50

Distributions In a continuous uniform distribution,


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a. Find the mean and standard deviation for the distribution of the waiting times represented in Figure 6-2, which accompanies Exercises 5–8.

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Identify the minimum and maximum values of the waiting times from the given distribution or figure. These values are necessary to calculate both the mean (μ) and the standard deviation (σ).
Use the formula for the mean of a continuous uniform distribution: μ = (minimum + maximum) / 2. Substitute the identified minimum and maximum values into this formula.
Calculate the range of the distribution, which is the difference between the maximum and minimum values: range = maximum - minimum.
Use the formula for the standard deviation of a continuous uniform distribution: σ = range / √12. Substitute the calculated range into this formula.
Simplify the expressions for both the mean and standard deviation to obtain the final results. Ensure all calculations are consistent with the provided formulas.

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Key Concepts

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

Continuous Uniform Distribution

A continuous uniform distribution is a probability distribution where all outcomes are equally likely within a specified range. This means that any value between the minimum and maximum is equally probable, leading to a flat probability density function. The distribution is defined by its two parameters: the minimum and maximum values, which determine the range of possible outcomes.
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Mean of a Uniform Distribution

The mean (μ) of a continuous uniform distribution is calculated as the average of the minimum and maximum values. It represents the central point of the distribution and is given by the formula μ = (minimum + maximum) / 2. This value indicates where the center of the distribution lies, providing insight into the expected value of a random variable drawn from this distribution.
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Standard Deviation of a Uniform Distribution

The standard deviation (σ) of a continuous uniform distribution measures the spread of the distribution around the mean. It is calculated using the formula σ = range / √12, where the range is the difference between the maximum and minimum values. A larger standard deviation indicates a wider spread of values, while a smaller standard deviation suggests that the values are closer to the mean.
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Related Practice
Textbook Question

Standard Normal Distribution. In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.


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Textbook Question

Continuous Uniform Distribution. In Exercises 5–8, refer to the continuous uniform distribution depicted in Figure 6-2 and described in Example 1. Assume that a passenger is randomly selected, and find the probability that the waiting time is within the given range.

Less than 4.00 minutes

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Textbook Question

Births: Sampling Distribution of Sample Proportion When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg (where and Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion?

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Textbook Question

Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.


Between 1.50 and 2.00

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Textbook Question

Constructing Normal Quantile Plots. In Exercises 17–20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution.


Earthquake Depths A sample of depths (km) of earthquakes is obtained from Data Set 24 “Earthquakes” in Appendix B: 17.3, 7.0, 7.0, 7.0, 8.1, 6.8.

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Textbook Question

Finding Bone Density Scores. In Exercises 37–40 assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the bone density test score corresponding to the given information. Round results to two decimal places.


Find P99, the 99th percentile. This is the bone density score separating the bottom 99% from the top 1%.

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