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Answer:To determine which statement is true regarding Allen's reading habits compared to the survey statistics, we need to consider his reading in the context of the mean and standard deviation provided.
Given information:
- Mean number of books read by adults in the survey: \( \mu = 12 \)
- Standard deviation of the number of books read: \( \sigma = 4 \)
- Allen read 20 books in the past year.
We want to compare Allen's reading of 20 books with the average and standard deviation of the survey.
1. **Z-score Calculation:**
The Z-score measures how many standard deviations Allen's reading of 20 books is away from the mean.
\[
Z = \frac{X - \mu}{\sigma}
\]
where:
- \( X \) is Allen's number of books read (20),
- \( \mu \) is the mean number of books read (12),
- \( \sigma \) is the standard deviation (4).
Calculate \( Z \):
\[
Z = \frac{20 - 12}{4} = \frac{8}{4} = 2
\]
Allen's Z-score is 2.
2. **Interpreting the Z-score:**
- A Z-score of 2 means that Allen's number of books read (20) is 2 standard deviations above the mean of the survey.
3. **Comparison and Conclusion:**
- Allen's Z-score of 2 indicates that he read significantly more books than the average respondent in the survey.
- This means Allen is in the top 2.5% of readers (since a Z-score of 2 corresponds to approximately the 97.5th percentile in a standard normal distribution).
Therefore, the true statement regarding Allen's reading compared to the survey statistics is that **Allen reads significantly more books than the average adult in the survey**. Specifically, he is in the top percentile of readers based on the Z-score calculation.
Step-by-step explanation:Certainly! Let's go through the steps to analyze Allen's reading habits compared to the survey statistics step-by-step.
Given information:
- Mean number of books read by adults in the survey: \( \mu = 12 \)
- Standard deviation of the number of books read: \( \sigma = 4 \)
- Allen read 20 books in the past year.
### Step 1: Calculate Allen's Z-score
The Z-score measures how many standard deviations Allen's reading of 20 books is away from the mean of the survey.
The formula for calculating the Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
where:
- \( X \) is the value we are comparing (Allen's number of books read, which is 20),
- \( \mu \) is the mean of the population (survey mean, which is 12),
- \( \sigma \) is the standard deviation of the population (survey standard deviation, which is 4).
Substitute the given values:
\[ Z = \frac{20 - 12}{4} \]
Calculate the numerator:
\[ 20 - 12 = 8 \]
Calculate the Z-score:
\[ Z = \frac{8}{4} = 2 \]
### Step 2: Interpret the Z-score
A Z-score of 2 means that Allen's number of books read (20) is 2 standard deviations above the mean of the survey.
### Step 3: Compare and Conclude
- A Z-score of 2 indicates that Allen read significantly more books than the average adult in the survey.
- Specifically, a Z-score of 2 corresponds to being in approximately the 97.5th percentile in a standard normal distribution.
- This means Allen is in the top 2.5% of readers based on the survey statistics.
### Conclusion
Therefore, the step-by-step analysis shows that **Allen reads significantly more books than the average adult in the survey**, placing him in the top percentile of readers based on the Z-score calculation of 2.
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