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Chapter 15: Problem 20

Mrs. Gillis gave a test to her two classes of algebra. The mean grade for her class of 20 students was 86 and the mean grade of her class of 15 students was \(79 .\) What is the mean grade when she combines the grades of both classes?

Short Answer

Expert verified
The combined mean grade is 83.

Step by step solution

01

Calculate the total sum of grades for each class

For the first class, multiply the mean by the number of students: \(20 \times 86 = 1720\). For the second class, multiply the mean by the number of students: \(15 \times 79 = 1185\).
02

Combine the total sums from both classes

Add the total sums from both classes: \(1720 + 1185 = 2905\).
03

Calculate the total number of students

Add the number of students from both classes: \(20 + 15 = 35\).
04

Calculate the combined mean grade

Divide the combined total sum of grades by the total number of students: \(\frac{2905}{35} = 83\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weighted Average
The concept of a weighted average is crucial when dealing with different group sizes. Here, the grades from two algebra classes are combined. Each class contributes to the overall average based on its size. Calculate each class's total grades first. For a weighted average:
  • Multiply each group's mean by its size.
  • Add these results together for the total grade sum.
  • Divide by the total number of students to find the overall mean.
This way, we account for the fact that the classes have different numbers of students, providing a more accurate combined average grade. In this case, the weighted average ensures that both classes' performances are fairly represented.
Algebra
Algebra often involves manipulating numbers and variables to solve problems. In this exercise, algebraic concepts help compute the combined average grade from two classes. The key steps involve using algebraic expressions:
  • Expression for each class's total grade: Class size multiplied by mean.
  • Total grade for both classes: Sum of individual class totals.
  • Total number of students: Sum of class sizes.
These straightforward algebraic steps showcase how combining information can solve real-world problems, such as determining grades or averages.
Arithmetic Mean
The arithmetic mean refers to the average of a set of numbers. It's a fundamental concept in statistics and mathematics, typically used to summarize data with a single value. For the arithmetic mean, follow these steps:
  • Add all numbers in the set together.
  • Divide the total by the count of numbers in the set.
In the exercise, the arithmetic mean is crucial for calculating both individual class averages and the combined mean. While each class has its distinct average, they collectively form a larger data set, allowing us to use the same arithmetic process to understand overall performance. This provides insight into how each group and the total contribute to the mean.

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Most popular questions from this chapter

In \(9-13 :\) a. Create a scatter plot for the data. b. Determine which regression model is the most appropriate for the data. Justify your answer. c. Find the regression equation. Round the coefficient of the regression equation to three decimal places. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline x & {4} & {7} & {3} & {8} & {6} & {5} & {6} & {3} & {9} & {4.5} \\ \hline y & {10} & {7} & {15} & {9} & {5} & {6} & {6} & {14} & {14} & {8} \\ \hline\end{array} $$

In \(11-14,\) select the numeral that precedes the choice that best completes the statement or answers the question. The heights of 200 women are normally distributed. The mean height is 170 centimeters with a standard deviation of 10 centimeters. What is the best estimate of the number of women in this group who are between 160 and 170 centimeters tall? $$ \begin{array}{llll}{\text { (1) } 20} & {\text { (2) } 34} & {\text { (3) } 68} & {\text { (4) } 136}\end{array} $$

In order to improve customer relations, an auto-insurance company surveyed 100 people to determine the length of time needed to complete a report form following an auto accident. The result of the survey is summarized in the following table showing the number of minutes needed to complete the form. Find the mean and median amount of time needed to complete the form. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline \text { Minutes } & {26-30} & {31-35} & {36-40} & {41-45} & {46-50} & {51-55} & {56-60} & {61-65} & {66-70} \\\ \hline \text { Frequency } & {2} & {8} & {12} & {15} & {10} & {24} & {26} & {1} & {2} \\ \hline\end{array} $$

In \(7-9,\) find the mean, median, range, and interquartile range for each set of data to the nearest tenth. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 11 & {5} \\ {16} & {8} \\\ {19} & {9} \\ {31} & {6} \\ {37} & {5} \\ {32} & {5} \\ {35} & {6} \\\ \hline\end{array} $$

A student's scores on five tests were \(98,97,95,93,\) and \(67 .\) Explain why this set of scores does not represent a normal distribution.
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