https://r2.erweima.ai/mathbot/image/1720740262095vqlb6d76.jpg

To solve question number 3, which involves computing the mean, median, and mode for the given data set of telephone call frequencies, let's analyze the data step by step. The data provided is as follows:

**Number of calls and their respective frequencies:**
```
No. of calls:    0   1   2   3   4   5   6
Frequency:      15  22  28  35  42  34  24
```

### 1. Mean
The mean is calculated as the sum of all values divided by the number of values. For grouped data, the mean can be found using the formula:
\[ \text{Mean} = \frac{\sum f_i x_i}{\sum f_i} \]
where \( f_i \) is the frequency of each class and \( x_i \) is the mid-point of each class.

First, we calculate the total frequency (\(\sum f_i\)):
\[ \sum f_i = 15 + 22 + 28 + 35 + 42 + 34 + 24 = 200 \]

Next, we calculate the sum of the products of each number of calls and their frequencies (\(\sum f_i x_i\)):
\[ \sum f_i x_i = (0 \times 15) + (1 \times 22) + (2 \times 28) + (3 \times 35) + (4 \times 42) + (5 \times 34) + (6 \times 24) \]
\[ \sum f_i x_i = 0 + 22 + 56 + 105 + 168 + 170 + 144 = 665 \]

So, the mean is:
\[ \text{Mean} = \frac{665}{200} = 3.325 \]

### 2. Median
To find the median, we need to determine the class in which the median lies. The median class is the one where the cumulative frequency is at least half the total frequency (\(\frac{N}{2}\)), where \( N \) is the total frequency.

\[ N = 200 \]
\[ \frac{N}{2} = \frac{200}{2} = 100 \]

We then find the cumulative frequencies:

\[
\begin{aligned}
&0-15\\
&1-15+22=37\\
&2-37+28=65\\
&3-65+35=100\\
&4-100+42=142\\
&5-142+34=176\\
&6-176+24=200\\
\end{aligned}
\]

The cumulative frequency just reaches or exceeds 100 at the class interval for 3 calls.

Using the formula for the median:
\[ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h \]
where:
- \( L \) = lower boundary of the median class = 3
- \( CF \) = cumulative frequency of the class before the median class = 65
- \( f \) = frequency of the median class = 35
- \( h \) = class width (since data is individual numbers, h = 1)

\[ \text{Median} = 3 + \left( \frac{100 - 65}{35} \right) \times 1 = 3 + \frac{35}{35} = 3 + 1 = 4 \]

### 3. Mode
The mode is the value that appears most frequently in the data set. In this case, the mode is simply the number of calls with the highest frequency.

The highest frequency is 42, which corresponds to 4 calls.

So, the mode is:
\[ \text{Mode} = 4 \]

### Summary
- Mean = 3.325
- Median = 3
- Mode = 4

Do you want details or have any questions? Here are five additional questions you can ask:
1. How do you calculate the standard deviation for this data set?
2. What is the probability distribution of the number of calls?
3. How can we find the range of the data set?
4. What is the interquartile range for this data set?
5. How do we perform a hypothesis test on this data set?

Tip: Understanding the distribution of data is crucial for making informed decisions based on statistical analysis.

Learn how to calculate the mean, median, and mode for a data set of telephone call frequencies with a detailed step-by-step solution. This guide covers basic statistical concepts and formulas suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation