Question

Estimate the mean for the distribution shown. State the estimated mean and explain why your answer is reasonable.


21 (2)
24 (2)
27 (4)
30 (8)
33 (12)
36 (16)
39 (16)
42 (18)
45 (16)
48 (14)

Answer 1

Are you sure

Answer 2

To estimate the mean for the distribution shown, we can use the formula for the weighted mean, where we take each value and multiply it by its corresponding frequency, sum those products, and then divide by the total frequency.

Here's the breakdown of the calculation:

1. **List Values and Frequencies:**
- \(21\) with frequency \(2\)
- \(24\) with frequency \(2\)
- \(27\) with frequency \(4\)
- \(30\) with frequency \(8\)
- \(33\) with frequency \(12\)
- \(36\) with frequency \(16\)
- \(39\) with frequency \(16\)
- \(42\) with frequency \(18\)
- \(45\) with frequency \(16\)
- \(48\) with frequency \(14\)

2. **Calculate Total Frequency (N):**
\[
N = 2 + 2 + 4 + 8 + 12 + 16 + 16 + 18 + 16 + 14 = 98
\]

3. **Calculate Weighted Sum of Values (Σf*x):**
\[
\begin{align*}
\text{For } 21: & \quad 21 \times 2 = 42 \\
\text{For } 24: & \quad 24 \times 2 = 48 \\
\text{For } 27: & \quad 27 \times 4 = 108 \\
\text{For } 30: & \quad 30 \times 8 = 240 \\
\text{For } 33: & \quad 33 \times 12 = 396 \\
\text{For } 36: & \quad 36 \times 16 = 576 \\
\text{For } 39: & \quad 39 \times 16 = 624 \\
\text{For } 42: & \quad 42 \times 18 = 756 \\
\text{For } 45: & \quad 45 \times 16 = 720 \\
\text{For } 48: & \quad 48 \times 14 = 672 \\
\end{align*}
\]
Now, we can sum these products:

\[
\text{Total weighted sum} = 42 + 48 + 108 + 240 + 396 + 576 + 624 + 756 + 720 + 672 = 3392
\]

4. **Calculate Estimated Mean:**
\[
\text{Mean} = \frac{\text{Total weighted sum}}{N} = \frac{3392}{98} \approx 34.63
\]

### Estimated Mean:
The estimated mean is approximately **34.63**.

### Explanation of Reasonability:
This estimated mean value falls within the range of the distribution provided (ranging from 21 to 48). The number of occurrences at each value indicates that values around 36 to 42 are occurring more frequently. The mean accounts for these frequencies; hence, our estimated mean of approximately 34.63 is reasonable given that it reflects a balanced average of the data, taking into account the skewness towards the higher frequencies of values between 36 and 42.