/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 In an examination given to a cla... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 39

In an examination given to a class of 20 students, the following test scores were obtained: $$\begin{array}{lllllllllr}40 & 45 & 50 & 50 & 55 & 60 & 60 & 75 & 75 & 80 \\ 80 & 85 & 85 & 85 & 85 & 90 & 90 & 95 & 95 & 100\end{array}$$ a. Find the mean (or average) score, the mode, and the median score. b. Which of these three measures of central tendency do you think is the least representative of the set of scores?

Short Answer

Expert verified
The mean score is 79.5, the mode is 85, and the median is 80. The least representative measure of central tendency is the mode (85) since it appears only four times and does not account for the other scores in the dataset.

Step by step solution

01

Calculate the Mean (Average) Score

To find the mean score, we add up all the scores and then divide by the total number of scores (20 in this case). Mean = \(\frac{Sum \: of \: scores}{Number \: of \: scores}\)
02

Calculate the Sum of Scores

Here's the sum of all the scores: Sum = 40 + 45 + 50 + 50 + 55 + 60 + 60 + 75 + 75 + 80 + 80 + 85 + 85 + 85 + 85 + 90 + 90 + 95 + 95 + 100 Sum = 1590
03

Calculate the Mean

Now, we calculate the mean by dividing the sum by the total number of scores: Mean = \(\frac{1590}{20}\) Mean = 79.5
04

Find the Mode

The mode is the number that appears most frequently in the dataset. In this case, 85 appears four times, which is the highest frequency, so: Mode = 85
05

Find the Median

To find the median, first, we order the scores in ascending order (which they already are), and then we find the middle value. Since there are 20 scores (an even number of scores), we take the mean of the two middle values (10th and 11th scores): Median = \(\frac{80+80}{2}\) Median = 80
06

Analyze the Measures of Central Tendency

We have calculated the mean (79.5), mode (85), and median (80). Now, we need to discuss which of these measures is the least representative of the set of scores. The mode (85) appears only four times out of the 20 scores, so it may not represent the scores' overall distribution well. The median (80) is the middle value, and it sits well within the range of values, so it could generally represent the scores. The mean (79.5) takes into account every score, but some extreme values such as low scores (40 and 45) and high scores (95 and 100) may not represent the overall group well.
07

Conclusion

Considering the mean, mode, and median, the least representative measure of central tendency for this dataset is the mode (85) because it appears only a few times and does not account for the other scores in the dataset.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mean
Understanding the concept of the "Mean" is crucial for interpreting data sets. The mean, often referred to as the average, is calculated by adding up all the values in a data set and then dividing by the number of values. It offers insights into the overall level of all the individual values combined.

For example, in our examination data, the mean score is computed by totaling all the scores and dividing by how many scores there are. By doing so, you arrive at a mean score of 79.5.
  • This mean takes every score into account.
  • Extremes, such as the lowest and highest scores, can skew this average.
The mean provides a middle ground, but it's essential to be cautious about how outliers (very high or very low scores) might impact the representation of a group.
Median
The "Median" is a measure of central tendency that identifies the middle point of a data set. It's found by listing numbers in order and locating the midpoint. If the data set has an even number of observations, the median is the mean of the two central numbers.

In the exam scores given, the scores are already sorted. With 20 scores, the median is the average of the 10th and 11th scores in this ordered list. Both are 80, making the median 80.
  • The median is not affected by extremely high or low values.
  • It provides a central point that splits the data into two equal halves.
Hence, the median is a reliable indicator of central tendency, especially when dealing with skewed distributions.
Mode
The "Mode" is the value that appears most frequently in a data set. It reflects the most common point or frequency within data, making it straightforward but unique among measures of central tendency.

In the situation of these exam scores, the number 85 appears more often than any other score (four times), making it the mode.
  • Mode is excellent for understanding the most frequent data point.
  • It may not always reflect the center of a data set if it occurs infrequently.
Thus, while the mode can show popular scores, it's less effective if the most common score doesn't form a significant part of the data set, as seen in the example where it isn't very spread out.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A Christmas tree light has an expected life of \(200 \mathrm{hr}\) and a standard deviation of \(2 \mathrm{hr}\). a. Find a bound on the probability that one of these Christmas tree lights will last between \(190 \mathrm{hr}\) and \(210 \mathrm{hr}\). b. Suppose 150,000 of these Christmas tree lights are used by a large city as part of its Christmas decorations. Estimate the number of lights that are likely to require replacement between \(180 \mathrm{hr}\) and \(220 \mathrm{hr}\) of use.

If a sample of three batteries is selected from a lot of ten, of which two are defective, what is the expected number of defective batteries?

Sally Leonard, a real estate broker. is relocating in a large metropolitan area where she has received job offers from realty company A and realty company B. The number of houses she expects to sell in a year at each firm and the associated probabilities are shown in the following tables. The average price of a house in the locale of company \(\mathrm{A}\) is $$\$ 308,000$$, whereas the average price of a house in the locale of company \(\mathrm{B}\) is $$\$ 474,000$$. If Sally will receive a \(3 \%\) commission on sales at both companies, which job offer should she accept to maximize her expected yearly commission?

A panel of 64 economists was asked to predict the average unemployment rate for the upcoming year. The results of the survey follow: $$\begin{array}{lccccccc} \hline \text { Unemployment } & & & & & & & \\ \text { Rate, \% } & 4.5 & 4.6 & 4.7 & 4.8 & 4.9 & 5.0 & 5.1 \\ \hline \text { Economists } & 2 & 4 & 8 & 20 & 14 & 12 & 4 \\ \hline\end{array}$$ Based on this survey, what does the panel expect the average unemployment rate to be next year?

Bob, the proprietor of Midway Lumber, bases his projections for the annual revenues of the company on the performance of the housing market. He rates the performance of the market as very strong, strong, normal, weak, or very weak. For the next year, Bob estimates that the probabilities for these outcomes are $$18, .27$$, $$.42, .10$$, and $$.03$$, respectively. He also thinks that the revenues corresponding to these outcomes are $$\$ 20, \$$ 18.8$$, $$\$ 16.2, \$$ 14$$, and $$\$ 12$$ million, respectively. What is Bob's expected revenue for next year?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.