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Each year, taxpayers are able to contribute money to various charities via their IRS tax forms. The following list contains amounts donated via IRS tax forms by 8 taxpayers:
[tex]\[ \$5, 61, 34, 22, 49, 32, 8, 100 \][/tex]

\begin{tabular}{|c|c|c|}
\hline
\begin{tabular}{l}
(a) What is the median of this data set? If your answer is not \\
an integer, round your answer to one decimal place.
\end{tabular} & [tex]$\square$[/tex] & \\
\hline
\begin{tabular}{l}
(b) What is the mean of this data set? If your answer is not an \\
integer, round your answer to one decimal place.
\end{tabular} & [tex]$\square$[/tex] & \\
\hline
\begin{tabular}{l}
(c) How many modes does the data set have, and what are \\
their values? Indicate the number of modes by clicking in the \\
appropriate circle, and then indicate the value(s) of the \\
mode(s), if applicable.
\end{tabular} & \begin{tabular}{l}
○ zero modes \\
○ one mode: \\
○ two modes:
\end{tabular} & \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
Certainly! Let's break down the solution step by step:

### Part (a): Finding the Median
The median is the middle value of a data set when it is ordered in increasing order. If the data set has an even number of observations, the median is the average of the two middle numbers.

First, let’s sort the data:
[tex]\[ 5, 8, 22, 32, 34, 49, 61, 100 \][/tex]

Since there are 8 (even number) observations, the median will be the average of the 4th and 5th numbers.
[tex]\[ \text{Median} = \frac{32 + 34}{2} = 33.0 \][/tex]

So, the median of this data set is [tex]\(33.0\)[/tex].

### Part (b): Finding the Mean
The mean is the sum of all observations divided by the number of observations.

[tex]\[ \text{Mean} = \frac{5 + 61 + 34 + 22 + 49 + 32 + 8 + 100}{8} = \frac{311}{8} = 38.875 \][/tex]

So, the mean of this data set is [tex]\(38.875\)[/tex].

### Part (c): Finding the Mode
The mode is the value that appears most frequently in the data set. A data set can have no mode, one mode, or multiple modes.

In this particular data set, each value appears exactly once:
[tex]\[ 5, 61, 34, 22, 49, 32, 8, 100 \][/tex]

Since no value repeats, the data set technically has many values tying as modes:
[tex]\[ 5, 8, 22, 32, 34, 49, 61, 100 \][/tex]

Thus, there are 8 modes because each number appears once.

### Summary in Table Form
[tex]\[ \begin{tabular}{|c|c|c|} \hline (a) & 33.0 & \\ \hline (b) & 38.875 & \\ \hline (c) & \begin{tabular}{c} zero modes \\ one mode: \\ two modes: \\ \end{tabular} & \text{All values are modes: 5, 8, 22, 32, 34, 49, 61, 100} \\ \hline \end{tabular} \][/tex]

So the final answers are:
- (a) Median: [tex]\(33.0\)[/tex]
- (b) Mean: [tex]\(38.875\)[/tex]
- (c) The data set has [tex]\(8\)[/tex] modes: [tex]\(5, 8, 22, 32, 34, 49, 61, 100\)[/tex]
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