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

Mean

Statistics

3. Describing Data Numerically

Mean

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3:01 minutes

Problem 2.3.50

Textbook Question

Finding the Mean of a Frequency Distribution In Exercises 49–52, approximate the mean of the frequency distribution.

Social Media The average daily amounts of time (in minutes) spent on Snapchat

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Step 1: Identify the midpoints of each class interval. The midpoint is calculated as the average of the lower and upper boundaries of each interval. For example, for the interval 0–19, the midpoint is (0 + 19) / 2 = 9.5.

Step 2: Multiply each midpoint by its corresponding frequency to find the 'frequency × midpoint' product for each class interval. For example, for the interval 0–19, the product is 9.5 × 8 = 76.

Step 3: Sum all the 'frequency × midpoint' products obtained in Step 2. This gives the total of all weighted midpoints.

Step 4: Calculate the total frequency by summing all the frequencies provided in the table. For example, the total frequency is 8 + 8 + 15 + 10 + 7 = 48.

Step 5: Divide the sum of 'frequency × midpoint' products (from Step 3) by the total frequency (from Step 4) to approximate the mean of the frequency distribution. The formula is Mean = (Σ(frequency × midpoint)) / (Σ(frequency)).

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Key Concepts

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

Mean of a Frequency Distribution

The mean of a frequency distribution is calculated by multiplying each class midpoint by its corresponding frequency, summing these products, and then dividing by the total number of observations. This provides a weighted average that reflects the distribution of data across different intervals.

Class Midpoint

The class midpoint is the value that lies in the middle of a class interval. It is calculated by averaging the lower and upper boundaries of the interval. For example, for the interval 0-19, the midpoint is (0 + 19) / 2 = 9.5, which is used in calculating the mean.

Frequency

Frequency refers to the number of occurrences of a particular value or range of values in a dataset. In the context of a frequency distribution, it indicates how many data points fall within each specified interval, which is essential for calculating measures like the mean.

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Textbook Question

For the four test scores 96, 85, 91, and 86, the first 3 test scores are 20% of the final grade, and the last test score is 40% of the final grade. Find the weighted mean of the test scores.

Textbook Question

According to data from the city of Toronto, Ontario, Canada, there were nearly 112,000 parking infractions in the city for December 2020, with fines totaling over 5,500,000 Canadian dollars. The fines (in Canadian dollars) for a random sample of 105 parking infractions in Toronto, Ontario, Canada, for December 2020 are listed below. (Source: City of Toronto)

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Find the sample mean of the data.

Textbook Question

Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.

Grades A student receives the grades shown below, with an A worth 4 points, a B worth 3 points, a C worth 2 points, and a D worth 1 point. What is the student’s grade point average?

Textbook Question

Grades In Exercise 46, one of the student’s B grades gets changed to an A. What is the student’s new grade point average?

Textbook Question

Finding the Mean of a Frequency Distribution In Exercises 49–52, approximate the mean of the frequency distribution.

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Textbook Question

Extending Concepts

Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.

a. Find the 10% trimmed mean for the data in Exercise 65.

Textbook Question

Extending Concepts

Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.

b. Compare the four measures of central tendency, including the midrange.

Textbook Question

Extending Concepts

Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.

c. What is the benefit of using a trimmed mean versus using a mean found using all data entries? Explain your reasoning.

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