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Chapter 15: Problem 7

In \(3-8,\) find the mean, the median, and the mode for each set of data. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline \$ 1.10 & {1} \\ {\$ 1.20} & {5} \\ {\$ 1.30} & {8} \\ {\$ 1.40} & {6} \\ {\$ 1.50} & {6} \\\ \hline\end{array} $$

Short Answer

Expert verified
Mean = 1.342, Median = 1.30, Mode = 1.30.

Step by step solution

01

Understand the Data

The set of data is given in a frequency distribution table. The first column represents the data points \(x_i\) and the second column \(f_i\) represents their respective frequencies. Thus, the data points are 1.10, 1.20, 1.30, 1.40, and 1.50 with frequencies 1, 5, 8, 6, and 6, respectively.
02

Calculate the Mean

To find the mean, first multiply each data point by its frequency and then sum these products. Divide this total by the sum of the frequencies. Calculate: \( (1.10 \times 1) + (1.20 \times 5) + (1.30 \times 8) + (1.40 \times 6) + (1.50 \times 6) = 1.10 + 6.00 + 10.40 + 8.40 + 9.00 = 34.90 \). Total frequency: \( 1 + 5 + 8 + 6 + 6 = 26 \). The mean is: \( \frac{34.90}{26} \approx 1.342 \).
03

Calculate the Median

The median is the middle value of an ordered data set. With respect to frequencies, it divides the distribution into two equal parts. Arrange the data in ascending frequency order: 1.10 (1), 1.20 (5), 1.30 (8), 1.40 (6), 1.50 (6). The cumulative frequency reaches 13 at \(1.30\), which falls in middle as 26/2 = 13. Thus, the median is 1.30.
04

Determine the Mode

The mode is the data point with the highest frequency in the dataset. From the frequency distribution table, you can see that the highest frequency is 8, which corresponds to the data point 1.30. Therefore, the mode is 1.30.

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Key Concepts

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

Understanding Frequency Distribution
A frequency distribution is a way to organize data that shows the number of occurrences (or frequency) of different values in a dataset. In simpler terms, it tells us how often certain values appear. In the given exercise, the data points like 1.10, 1.20, 1.30, etc., are paired with numbers that tell us how frequently each amount occurs. For example, the number 1.20 appears 5 times in the dataset.
By understanding this concept, you can easily visualize how the data is spread across different values. This helps in further calculations like mean, median, and mode by simplifying the process.
Exploring Central Tendency
Central tendency refers to a central value or a typical size for a set of data. It is a measure that represents the center point of a dataset and includes mean, median, and mode.
  • The mean provides an average, considering all data points.
  • The median indicates the middle value that divides the dataset into two equal halves.
  • The mode is the most frequently occurring value.

Each has its own significance and use, depending on the data structure and the kind of analysis we wish to perform.
Calculating Mean
The mean is an average of all data points, giving a sense of the overall level of values in the dataset. To find it, you multiply each data point by its frequency (which reflects how many times it occurs) and sum all these products. Afterward, divide by the total number of data points to find the average.
In our dataset, mean calculation involved multiplying each value by its corresponding frequency (like 1.10 multiplied by 1), then adding them all up to get 34.90. Dividing this sum by the total frequency (26) gives us the mean, approximately 1.342.
Overall, the mean offers a great way to understand the general tendency of the data.
Determining Median
The median is the value that lies at the midpoint of the dataset. It divides the data into two equal parts. To find the median in a frequency distribution, the data must first be ordered.
In our frequency table, when we accumulate frequencies until we reach or exceed halfway through the total frequency count (which is 13 in this exercise), the corresponding data point is the median. Here, as the cumulative frequency reaches 13 with the value 1.30, this number becomes the median.
The median is pivotal when assessing data that might have outliers or skewed distributions, as it is less affected by extreme values than the mean.
Finding Mode
The mode is simple yet powerful, representing the value that appears most frequently in the dataset. To identify the mode in a frequency distribution, look at which data point has the highest frequency.
From our example, the data point 1.30 appears most often (8 times), making it the mode. Unlike the mean and median, a dataset can have more than one mode if multiple values share the highest frequency, or no mode if all frequencies are equal.
The mode is particularly useful for categorical data where mean and median cannot be defined, and it gives insight into the most common value within a dataset.

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Most popular questions from this chapter

In \(11-17 :\) a. Draw a scatter plot. b. Does the data set show strong positive linear correlation, moderate positive linear correlation, no linear correlation, moderate negative linear correlation, or strong negative linear correlation? c. If there is strong or moderate correlation, write the equation of the regression line that approximates the data. The following table shows the number of gallons of gasoline needed to fill the tank of a car and the number of miles driven since the previous time the tank was filled. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Gallons } & {8.5} & {7.6} & {9.4} & {8.3} & {10.5} & {8.7} & {9.6} & {4.3} & {6.1} & {7.8} \\ \hline \text { Miles } & {255} & {230} & {295} & {250} & {315} & {260} & {290} & {130} & {180} & {235} \\ \hline\end{array} $$

The table shows the number of correct answers on a test consisting of 15 questions. The table represents correct answers for a sample of the students who took the test. Find the standard deviation based on this sample. \(\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Correct } & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} & {14} & {15} \\ \hline \text { Frequency } & {2} & {1} & {3} & {3} & {5} & {8} & {8} & {5} & {4} & {1} \\\ \hline\end{array}\)

In \(3-8,\) find the mean, the median, and the mode for each set of data. $$ \begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 50 & {8} \\ {40} & {12} \\\ {30} & {17} \\ {20} & {10} \\ {10} & {3} \\ \hline\end{array} $$

In \(11-14,\) select the numeral that precedes the choice that best completes the statement or answers the question. The playing life of a Euclid mp3 player is normally distributed with a mean of \(30,000\) hours and a standard deviation of 500 hours. Matt's mp3 player lasted for \(31,500\) hours. His \(\mathrm{mp} 3\) player lasted longer than what percent of other Euclid mp3 players? $$ \begin{array}{llll}{\text { (1) } 68 \%} & {\text { (2) } 95 \%} & {\text { (3) } 99.7 \%} & {\text { (4) more than } 99.8 \%}\end{array} $$

In \(15-19\) a. Draw a scatter plot for each data set. Based on the scatter plot, would the correlation coefficient be close to \(-1,0,\) or 1\(?\) Explain. c. Use a calculator to find the correlation coefficient for each set of data. The table below shows the five-day forecast and the actual high temperature for the fifth day over the course of 18 days. The temperature is given in degrees Fahrenheit.
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