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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1:53 minutes

Problem 2.3.41

Textbook Question

Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.
Final Grade The scores and their percents of the final grade for a statistics student are shown below. What is the student’s mean score?

Verified step by step guidance

Step 1: Understand the concept of weighted mean. The weighted mean is calculated by multiplying each score by its corresponding weight (percentage of the final grade), summing these products, and then dividing by the total weight (which should sum to 100%).

Step 2: Convert the percentages into decimal form for easier calculations. For example, 5% becomes 0.05, 35% becomes 0.35, and so on.

Step 3: Multiply each score by its corresponding weight in decimal form. For example, for Homework: 85 × 0.05, for Quizzes: 80 × 0.35, for Project/Speech: 100 × 0.35, and for Final Exam: 93 × 0.25.

Step 4: Add all the weighted scores together. This sum represents the numerator of the weighted mean formula.

Step 5: Divide the sum of the weighted scores by the total weight (which is 1, since all percentages add up to 100%). This will give the weighted mean of the student's scores.

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Key Concepts

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

Weighted Mean

The weighted mean is a type of average that takes into account the relative importance of each value. Unlike a simple average, where all values contribute equally, the weighted mean assigns different weights to each value based on their significance, often represented as percentages. This is particularly useful in scenarios like calculating final grades, where different assessments contribute differently to the overall score.

Percentages

Percentages represent a fraction of 100 and are used to express how a part relates to a whole. In the context of the weighted mean, percentages indicate the weight or importance of each score in the overall calculation. Understanding how to convert these percentages into decimal form is crucial for accurately computing the weighted mean.

Calculation of Weighted Mean

To calculate the weighted mean, multiply each score by its corresponding weight (percentage), sum these products, and then divide by the total of the weights. This process ensures that each score contributes to the final mean in proportion to its significance, allowing for a more accurate representation of the student's overall performance based on the different assessments.

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Related Practice

Textbook Question

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

Populations The populations (in thousands) of the counties in Montana in 2019 (Source: U.S. Census Bureau)

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.

Textbook Question

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

Credit Card Balance For the month of October, a credit card has a balance of $115.63 for 12 days, $637.19 for 6 days, $1225.06 for 7 days, $0 for 2 days, and $34.88 for 4 days. What is the account’s mean daily balance for October?

Multiple Choice

Find the mean of the sample data below.

Textbook Question

Geometric Mean The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 0.58%, 0.29%, 0.13%, 0.14%, 0.15%, and 0.19%. Identify the single percentage growth rate that is the same as the six consecutive growth rates by computing the geometric mean of 1.0058, 1.0029, 1.0013, 1.0014, 1.0015, and 1.0019.

Textbook Question

Weighted Mean A student of the author earned grades of 63, 91, 88, 84, and 79 on her five regular statistics tests. She earned grades of 86 on the final exam and 90 on her class projects. Her combined homework grade was 70. The five regular tests count for 60% of the final grade, the final exam counts for 10%, the project counts for 15%, and homework counts for 15%. What is her weighted mean grade? What letter grade did she earn (A, B, C, D, or F)? Assume that a mean of 90 or above is an A, a mean of 80 to 89 is a B, and so on.

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