To find the estimated mean from a frequency distribution, we use the following formula:

$\hat{\mu} = \frac{\sum (f \times x)}{\sum f}$

Where:

• $f$ is the frequency of each class.
• $x$ is the midpoint of each class.

Step 1: Find the midpoints of each class.

• For the class 40−44: midpoint $x = \frac{40 + 44}{2} = 42$
• For the class 45−49: midpoint $x = \frac{45 + 49}{2} = 47$
• For the class 50−54: midpoint $x = \frac{50 + 54}{2} = 52$
• For the class 55−59: midpoint $x = \frac{55 + 59}{2} = 57$
• For the class 60−64: midpoint $x = \frac{60 + 64}{2} = 62$

Step 2: Multiply the frequency by the corresponding midpoint.

Class (°F)	Midpoint $x$	Frequency $f$	$f \times x$
40−44	42	22	$22 \times 42 = 924$
45−49	47	44	$44 \times 47 = 2068$
50−54	52	12	$12 \times 52 = 624$
55−59	57	5	$5 \times 57 = 285$
60−64	62	2	$2 \times 62 = 124$

Step 3: Sum the $f \times x$ and the frequencies.

$\sum (f \times x) = 924 + 2068 + 624 + 285 + 124 = 4025$ $\sum f = 22 + 44 + 12 + 5 + 2 = 85$

Step 4: Calculate the estimated mean.

$\hat{\mu} = \frac{4025}{85} \approx 47.4 \, \text{°F}$

Step 5: Interpretation of the Mean

The estimated mean is 47.4°F (rounded to the nearest tenth).

Part 2: Can we safely conclude that this is the actual mean?

The correct answer is C:

• No, because we do not know the actual data values. We can only use the class midpoints.

The estimated mean from a frequency distribution is based on class midpoints, not the exact data values. The actual mean could be slightly different due to the spread within each class.

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Would you like more details or have any questions? Here are 5 related questions you could explore:

1. What is the formula for calculating the median in a frequency distribution?
2. How does skewness affect the relationship between mean and median?
3. Can the mode be estimated from a frequency distribution?
4. What is the significance of using midpoints in a grouped frequency distribution?
5. How would the estimated mean change if the class intervals were narrower?

Tip: When working with grouped data, always remember that the estimated mean is an approximation and may not reflect the exact mean of the original dataset.