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Some researchers developing a new intelligence test are trying to decide how much time to allow to complete the test. The researchers have recorded the times (in minutes) for completion of 32 people who took the test for practice. The frequency distribution below summarizes the completion times recorded by the researchers.

[tex]\[
\begin{tabular}{|c|c|}
\hline
\text{Time for completion} & \text{Frequency} \\
\text{(in minutes)} & \\
\hline
9 to 11 & 8 \\
12 to 14 & 9 \\
15 to 17 & 7 \\
18 to 20 & 5 \\
21 to 23 & 3 \\
\hline
\end{tabular}
\][/tex]

Based on the frequency distribution, using the midpoint of each data class, estimate the mean completion time of the people who took the test. For your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

[tex]\[
\boxed{\text{minutes}}
\][/tex]



Answer :

GinnyAnswer
To estimate the mean completion time using the midpoints of each class interval in the frequency distribution provided, we'll follow these steps:

1. Identify the midpoints of each class interval:
- For the class interval 9 to 11: midpoint = [tex]\( \frac{9 + 11}{2} = 10 \)[/tex]
- For the class interval 12 to 14: midpoint = [tex]\( \frac{12 + 14}{2} = 13 \)[/tex]
- For the class interval 15 to 17: midpoint = [tex]\( \frac{15 + 17}{2} = 16 \)[/tex]
- For the class interval 18 to 20: midpoint = [tex]\( \frac{18 + 20}{2} = 19 \)[/tex]
- For the class interval 21 to 23: midpoint = [tex]\( \frac{21 + 23}{2} = 22 \)[/tex]

2. Multiply each midpoint by the corresponding frequency:
- For class 9 to 11: [tex]\( 10 \times 8 = 80 \)[/tex]
- For class 12 to 14: [tex]\( 13 \times 9 = 117 \)[/tex]
- For class 15 to 17: [tex]\( 16 \times 7 = 112 \)[/tex]
- For class 18 to 20: [tex]\( 19 \times 5 = 95 \)[/tex]
- For class 21 to 23: [tex]\( 22 \times 3 = 66 \)[/tex]

3. Sum these products to obtain the total sum of products:
[tex]\[ 80 + 117 + 112 + 95 + 66 = 470 \][/tex]

4. Sum the frequencies to find the total number of observations:
[tex]\[ 8 + 9 + 7 + 5 + 3 = 32 \][/tex]

5. Calculate the mean completion time:
- Sum of products = 470
- Total number of observations = 32

Mean completion time = [tex]\( \frac{470}{32} \approx 14.6875 \)[/tex]

6. Round the mean completion time to one decimal place:
[tex]\[ 14.7 \][/tex]

Thus, the estimated mean completion time of the people who took the test is [tex]\( \boxed{14.7} \)[/tex] minutes.

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