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Chapter 3: Problem 2

Compute the mean of the following population values: 7,5,7,3,7,4

Short Answer

Expert verified
The mean of the population values is 5.5.

Step by step solution

01

Understand the Mean

The mean is the average of a set of numbers. It is calculated by adding all the numbers together and then dividing by the number of numbers.
02

Count the Numbers

First, count how many numbers are in the given set. In this case, the numbers given are 7, 5, 7, 3, 7, and 4. There are 6 numbers in total.
03

Add the Numbers

Add all the numbers together: \(7 + 5 + 7 + 3 + 7 + 4\). The sum of these numbers is 33.
04

Divide by the Total Count

Divide the sum of the numbers by the total count of numbers to find the mean: \(\frac{33}{6}\). This division gives 5.5.

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Key Concepts

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

Population Mean
The concept of "Population Mean" is important in statistics, particularly when analyzing a complete set of data. The population mean, often symbolized as \( \mu \), is the mean of all values in a population set.
It represents the central tendency or the average of all the data points in that specific set.
To calculate the population mean, you use the same method as you would when finding the mean for a sample. This involves a straightforward process:
  • Summing all the values in the data set.
  • Dividing the total by the number of data points in that population.

By doing this, the population mean provides insight into the overall level of the dataset, indicating where the central point or "average" lies.
Step-by-Step Guide
Understanding and calculating the mean can be greatly simplified through a step-by-step approach.
Here's a simple guide using the example dataset 7, 5, 7, 3, 7, 4:
1. **Count the numbers**: Begin by counting all the numbers in the set. Here, there are 6 numbers.2. **Add the numbers**: Sum up all the data values. When we add 7 + 5 + 7 + 3 + 7 + 4, we get 33.3. **Divide by the count**: Take the total sum (33) and divide it by the number of items (6). This calculation is \(\frac{33}{6}\).
This step-by-step breakdown simplifies the process and helps prevent common mistakes by focusing on the logical sequence of actions needed to find the mean.
Average Calculation
Calculating an average, also known as the mean, is a basic yet fundamental concept in statistics. It's an essential tool for understanding data and is often used to convey simplified conclusions.
The average is calculated by dividing the sum of all numbers in a dataset by how many numbers there are in that set.
  • For instance, with numbers 7, 5, 7, 3, 7, and 4:
  • First, add the values: \(7 + 5 + 7 + 3 + 7 + 4 = 33\).
  • Then, divide by 6: \(\frac{33}{6} = 5.5\).

The result, 5.5, is the mean of this dataset. This "average" value gives a sense of the typical number in a dataset and can provide a quick snapshot of that data set's tendencies. By summarizing data with just one number, the mean helps you understand and compare different sets of information easily.

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Most popular questions from this chapter

The accounting firm of Rowatti and Koppel specializes in income tax returns for self-employed professionals, such as physicians, dentists, architects, and lawyers. The firm employs 11 accountants who prepare the returns. For last year, the number of returns prepared by each accountant was: $$ \begin{array}{|lllllllllll|} \hline 58 & 75 & 31 & 58 & 46 & 65 & 60 & 71 & 45 & 58 & 80 \\ \hline \end{array} $$ Find the mean, median, and mode for the number of returns prepared by each accountant. If you could report only one, which measure of location would you recommend reporting?

The demand for the video games provided by Mid-Tech Video Games Inc. has exploded in the last several years. Hence, the owner needs to hire several new technical people to keep up with the demand. Mid-Tech gives each applicant a special test that Dr. McGraw, the designer of the test, believes is closely related to the ability to create video games. For the general population, the mean on this test is \(100 .\) Below are the scores on this test for the applicants. $$ \left.(\begin{array}{llllllllll} 95 & 105 & 120 & 81 & 90 & 115 & 99 & 100 & 130 & 10 \end{array}\right) $$ The owner is interested in the overall quality of the job applicants based on this test. Compute the mean and the median scores for the 10 applicants. What would you report to the owner? Does it seem that the applicants are better than the general population?

Ten young adults living in California rated the taste of a newly developed sushi pizza, topped with tuna, rice, and kelp, on a scale of 1 to \(50,\) with 1 indicating they did not like the taste and 50 that they did. The ratings were: $$ \begin{array}{lllllllll} 34 & 39 & 40 & 46 & 33 & 31 & 34 & 14 & 15 & 45 \end{array} $$ In a parallel study, 10 young adults in lowa rated the taste of the same pizza. The ratings were: $$ \begin{array}{lllllllll} 28 & 25 & 35 & 16 & 25 & 29 & 24 & 26 & 17 & 20 \end{array} $$ As a market researcher, compare the potential for sushi pizza in the two markets.

a. Compute the mean of the following sample values: 5,9,4,10 . b. Show that \(\Sigma(x-\bar{x})=0\).

Sally Reynolds sells real estate along the coastal area of Northern California. Below are her total annual commissions between 2007 and 2017 . Find the mean, median, and mode of the commissions she earned for the 11 years. $$ \begin{array}{|lc|} \hline \text { Year } & \text { Amount (thousands) } \\ \hline 2007 & 292.16 \\ 2008 & 233.80 \\ 2009 & 206.97 \\ 2010 & 202.67 \\ 2011 & 164.69 \\ 2012 & 206.53 \\ 2013 & 237.51 \\ 2014 & 225.57 \\ 2015 & 255.33 \\ 2016 & 248.14 \\ 2017 & 269.11 \\ \hline \end{array} $$
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