To evaluate whether a normal curve can be used to model the data using the Empirical Rule, we need to calculate the number of data points that fall within one standard deviation of the mean and compare it to the expected percentage of approximately 68% for a normal distribution.
Given:
- Mean (\( \mu \)) = 10.38
- Standard deviation (\( \sigma \)) = 0.80
- Data points: 8.70, 9.40, 9.70, 9.90, 10.00, 10.10, 10.30, 10.50, 10.50, 10.60, 10.90, 10.95, 11.10, 11.20, 11.90
First, we identify the range within one standard deviation of the mean:
- Lower bound: \( \mu - \sigma = 10.38 - 0.80 = 9.58 \)
- Upper bound: \( \mu + \sigma = 10.38 + 0.80 = 11.18 \)
Next, we count how many of the data points fall within this range (from 9.58 to 11.18):
- Data points within range:
- 9.70
- 9.90
- 10.00
- 10.10
- 10.30
- 10.50
- 10.50
- 10.60
- 10.90
- 10.95
- 11.10
- 11.20
Counting these, we find there are 12 data points within the range.
The total number of data points is 15.
To calculate the percentage of data points that fall within one standard deviation of the mean:
\[ \text{Percentage} = \left( \frac{12}{15} \right) \times 100 \approx 80% \]
Since 80% is much greater than the expected 68% for a normal distribution, it suggests that the distribution of the data may not be normal.
Thus, based on the options provided, the correct interpretation would be related to the high percentage of data within one standard deviation, suggesting a normal curve cannot be conclusively said to fit the data. But since 80% is not close to the expected percentage, you would likely choose an option that indicates that:
73 percent of the data falls within 1 standard deviation of the mean. A normal curve can be used to model the data, because this is approximately 68% of the data. (This option would be misleading since you have more than 68%, but it reflects the idea of closeness).
The correct reasoning based on the exact calculation you performed leads to the conclusion that 80% falls within that range, thus confirming that a normal distribution would not accurately describe the data based on common empirical guide alignments (like the empirical rule). Therefore, the option related to "55 percent" can be chosen if this were a simple categorical assessment.
However, if you are looking for the exact fit, no provided choice captures the 80% result as described, indicating that while one could argue a statistical clarity based on common rules, the specific dataset appears not to conform closely as expected.
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