Join IDNLearn.com and start exploring the answers to your most pressing questions. Get accurate and detailed answers to your questions from our dedicated community members who are always ready to help.

Find the mean of the data summarized in the given frequency distribution. Compare the computed mean to the actual mean of 56.4 degrees.

| Low Temperature (°F) | 40-44 | 45-49 | 50-54 | 55-59 | 60-64 |
|----------------------|-------|-------|-------|-------|-------|
| Frequency | 3 | 6 | 12 | 7 | 3 |

The mean of the frequency distribution is _____ degrees.
(Round to the nearest tenth as needed.)


Sagot :

To find the mean of the data summarized in the given frequency distribution, we will follow these steps:

1. Determine the midpoints for each interval:
- For the interval [tex]\(40-44\)[/tex], the midpoint is [tex]\((40 + 44) / 2 = 42\)[/tex].
- For the interval [tex]\(45-49\)[/tex], the midpoint is [tex]\((45 + 49) / 2 = 47\)[/tex].
- For the interval [tex]\(50-54\)[/tex], the midpoint is [tex]\((50 + 54) / 2 = 52\)[/tex].
- For the interval [tex]\(55-59\)[/tex], the midpoint is [tex]\((55 + 59) / 2 = 57\)[/tex].
- For the interval [tex]\(60-64\)[/tex], the midpoint is [tex]\((60 + 64) / 2 = 62\)[/tex].

2. List the midpoints and their corresponding frequencies:
[tex]\[ \begin{array}{cc} \text{Midpoint} & \text{Frequency} \\ \hline 42 & 3 \\ 47 & 6 \\ 52 & 12 \\ 57 & 7 \\ 62 & 3 \\ \end{array} \][/tex]

3. Calculate the weighted sum of the midpoints:
- Multiply each midpoint by its corresponding frequency and sum the results:
[tex]\[ (42 \times 3) + (47 \times 6) + (52 \times 12) + (57 \times 7) + (62 \times 3) \][/tex]
[tex]\[ = 126 + 282 + 624 + 399 + 186 \][/tex]
[tex]\[ = 1617 \][/tex]

4. Calculate the total number of data points:
- Sum the frequencies:
[tex]\[ 3 + 6 + 12 + 7 + 3 = 31 \][/tex]

5. Find the mean of the frequency distribution:
- Divide the weighted sum of the midpoints by the total number of data points:
[tex]\[ \text{Mean} = \frac{1617}{31} \approx 52.161 \][/tex]
- Rounded to the nearest tenth:
[tex]\[ \text{Mean} \approx 52.2 \][/tex]

So, the mean of the frequency distribution is approximately [tex]\(52.2\)[/tex] degrees.

When comparing this computed mean to the actual mean of [tex]\(56.4\)[/tex] degrees, we observe that the computed mean is lower. This indicates that the frequency distribution is somewhat skewed toward the lower intervals.