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 25m
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

3. Describing Data Numerically

Median

Statistics

3. Describing Data Numerically

Median

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1:33 minutes

Problem 2.3.3

Textbook Question

Building Basic Skills and Vocabulary

True or False? In Exercises 1–4, determine whether the statement is true or false. If it is false, rewrite it as a true statement.

A data set can have the same mean, median, and mode.

Verified step by step guidance

Understand the definitions of mean, median, and mode: The mean is the average of all data points, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value in the data set.

Recognize that it is possible for a data set to have the same mean, median, and mode. For example, in a perfectly symmetric distribution like a normal distribution, these three measures of central tendency can coincide.

Consider an example: A data set such as {3, 3, 3, 3, 3} has a mean of 3, a median of 3, and a mode of 3. This demonstrates that the statement can be true.

If the statement were false, you would need to provide a counterexample where the mean, median, and mode are not the same. For instance, in a skewed distribution, these values often differ.

Conclude that the statement is true because there are scenarios where the mean, median, and mode can indeed be the same, particularly in symmetric distributions or uniform data sets.

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Key Concepts

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

Mean

The mean is the average of a data set, calculated by summing all the values and dividing by the number of values. It is sensitive to extreme values, which can skew the mean significantly. In a perfectly symmetrical distribution, the mean represents the center of the data.

Median

The median is the middle value of a data set when the values are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle values. The median is less affected by outliers and provides a better measure of central tendency for skewed distributions.

Mode

The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values occur with the same frequency. The mode is particularly useful for categorical data where we wish to know the most common category.

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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

b. Find the median.

Textbook Question

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1.1 1.7 1.7 1.1 1.4 1.0 1.2 1.8 1.6 1.1

Textbook Question

Reported and Measured Heights Listed below are self-reported heights of males aged 16 and over and their corresponding measured heights (based on data from the National Health and Nutrition Examination Survey). All heights are in inches. First find the differences (reported height–measured height), and then use those differences to find the (a) mean, (b) median, (c) mode,

Textbook Question

Constructing Data Sets In Exercises 5– 8, construct the described data set. The entries in the data set cannot all be the same.

Mean and median are the same and the data is bimodal.

Textbook Question

Using and Interpreting Concepts

Finding and Discussing the Mean, Median, and Mode In Exercises 17–34, find the mean, the median, and the mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.

Cholesterol The cholesterol levels of a sample of 10 female employees

154 240 171 188 235 203 184 173 181 275

Textbook Question

Using and Interpreting Concepts

Judicial System The responses of a sample of 34 young adult United Kingdom males in custodial sentences who were asked what is affected by such sentences (Adapted from User Voice)

Mental health: 8

Trust: 3

Education: 8

Personal development: 5

Family: 3

Future opportunities: 3

Other: 4

Textbook Question

Using and Interpreting Concepts

Weights (in pounds) of Packages on a Delivery Truck

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