Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Probabilities & Z-Scores w/ Graphing Calculator

Statistics

6. Normal Distribution and Continuous Random Variables

Probabilities & Z-Scores w/ Graphing Calculator

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2:13 minutes

Problem 6.1.6

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

Verified step by step guidance

Step 1: Understand the continuous uniform distribution. In this case, the waiting time is uniformly distributed between 0 and 5 minutes, as shown in the graph. The probability density function (PDF) is constant at 0.2, and the total area under the curve equals 1.

Step 2: Recall the formula for the probability in a continuous uniform distribution. The probability of a random variable falling within a range [a, b] is given by the area under the curve between those limits. This is calculated as P(a ≤ X ≤ b) = (b - a) × f(x), where f(x) is the constant height of the PDF.

Step 3: Identify the range of interest. The problem asks for the probability that the waiting time is less than 4.00 minutes. This corresponds to the range [0, 4].

Step 4: Apply the formula. Substitute the values into the formula: P(0 ≤ X ≤ 4) = (4 - 0) × 0.2. This represents the area of the rectangle from x = 0 to x = 4 under the curve.

Step 5: Interpret the result. The calculated area represents the probability that the waiting time is less than 4.00 minutes. This probability is proportional to the area under the curve within the specified range.

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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. The probability density function (PDF) is constant across this range, resulting in a rectangular shape when graphed. For example, if waiting times are uniformly distributed between 0 and 5 minutes, every time within this interval has the same likelihood of occurring.

Probability Density Function (PDF)

The probability density function (PDF) describes the likelihood of a continuous random variable taking on a specific value. In the case of a continuous uniform distribution, the PDF is a horizontal line, indicating that the probability is evenly distributed across the range. The area under the PDF curve represents the total probability, which equals 1 for a valid distribution.

Calculating Probability

To find the probability of a continuous random variable falling within a certain range, you calculate the area under the PDF over that interval. For example, to find the probability that a passenger's waiting time is less than 4 minutes in a uniform distribution from 0 to 5 minutes, you would determine the area of the rectangle formed by the interval from 0 to 4, which is the height of the PDF multiplied by the width of the interval.

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Related Practice

Multiple Choice

Sketch a graph to represent the probability, then use a calculator to find it.

$P\left(Z>1.14\right)$

Multiple Choice

Use a calculator to find the z–scores of the region shown in the standard normal distribution below.

A = 0.800

Textbook Question

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

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.

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Greater than 3.00 minutes

Textbook Question

Between 2 minutes and 3 minutes

Textbook Question

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.

Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).

35 35 20 50 95 75 45 50 30 35 30 30

e. Convert the longest wait time to a z score.

f. Based on the result from part (e), is the longest wait time significantly high?

Textbook Question

Pulse Rates. In Exercises 13–24, use the data in the table below for pulse rates of adult males and females (based on Data Set 1 “Body Data” in Appendix B). Hint: Draw a graph in each case.

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Find the probability that a male has a pulse rate between 70 beats per minute and 90 beats per minute.

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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

Key Concepts

Continuous Uniform Distribution

Probability Density Function (PDF)

Calculating Probability

Watch next

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