Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

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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1:52 minutes

Problem 6.1.50

Textbook Question

Distributions In a continuous uniform distribution,

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

Verified step by step guidance

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.

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.

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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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

Less than 4.00 minutes

Textbook Question

Between 2 minutes and 3 minutes

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.

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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For males, find P90 which is the pulse rate separating the bottom 90% from the top 10%.

Textbook Question

z Scores. In Exercises 5–8, express all z scores with two decimal places.

Diastolic Blood Pressure of Females For the diastolic blood pressure measurements of females listed in Data Set 1 “Body Data” in Appendix B, the highest measurement is 98 mm Hg. The 147 diastolic blood pressure measurements of females have a mean of 70.2 mm Hg and a standard deviation of 11.2 mm Hg.

c. Convert the highest diastolic blood pressure to a z score.

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