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

2. Describing Data with Tables and Graphs

Frequency Distributions

Statistics

2. Describing Data with Tables and Graphs

Frequency Distributions

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2:24 minutes

Problem 2.1.1

Textbook Question

Boston Commute Time The accompanying table summarizes daily commute times in Boston. How many commute times are included in the summary? Is it possible to identify the exact values of all of the original data amounts?

Verified step by step guidance

Step 1: To determine the total number of commute times included in the summary, sum the frequencies provided in the table. The frequencies are: 488, 422, 92, 10, and 8.

Step 2: Add these frequencies together using the formula: \( \text{Total Commute Times} = \text{Frequency}_1 + \text{Frequency}_2 + \text{Frequency}_3 + \text{Frequency}_4 + \text{Frequency}_5 \).

Step 3: To address whether it is possible to identify the exact values of all original data amounts, consider the nature of the data. The table provides grouped data (intervals of commute times) rather than individual commute times. This means the exact values of the original data cannot be determined, only the range within which they fall.

Step 4: Understand that grouped data summarizes the frequency of values within intervals, but does not retain the specific individual data points. For example, in the interval 0–29 minutes, there are 488 commute times, but their exact values are unknown.

Step 5: Conclude that while the total number of commute times can be calculated, the exact values of all original data amounts cannot be identified from the given summary table.

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Key Concepts

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

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. In the context of the table, it shows the number of commuters falling within specific time ranges, allowing for a clear understanding of the distribution of commute times in Boston.

Cumulative Frequency

Cumulative frequency is the running total of frequencies through the classes in a frequency distribution. It helps in understanding how many observations fall below a particular value, which can be useful for identifying trends in commute times.

Data Reconstruction

Data reconstruction refers to the process of estimating or inferring the original data values from summarized data, such as frequency distributions. In this case, while we can determine the total number of commute times, we cannot identify the exact values of all original data amounts due to the grouped nature of the data.

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Related Practice

Multiple Choice

Use the frequency distribution below to find the class width and class midpoints.

Multiple Choice

The following data set shows the number of overtime hours that 12 employees worked in a month. Construct a frequency distribution, using a lower class limit of 3 and a class width of 4.

Multiple Choice

A data set has a minimum value of 16 and a maximum value of 71. Without constructing a table, find the class width if you organized this data into 7 classes. Write the lower and upper class limits.

min = 16; max = 71; 7 classes.

Textbook Question

In Exercises 5–8, identify the class width, class midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary. The frequency distributions are based on real data from Appendix B.

Textbook Question

In Exercises 9–12, using a loose interpretation of the criteria for determining whether a frequency distribution is approximately a normal distribution, determine whether the given frequency distribution is approximately a normal distribution. Give a brief explanation.

Best Actors Refer to the frequency distribution from Exercise 6

Textbook Question

Presidents Listed below are the ages (years) of presidents of the United States at the times of their first inaugurations (from Data Set 22 “Presidents” in Appendix B). Presidents who took office as a result of an assassination or resignation are not included. The data are current as of this writing. Use these ages to construct a frequency distribution. Use a class width of 5 years and begin with a lower class limit of 40 years. Do the ages appear to have a normal distribution?

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