To determine if a normal curve can be used to model the data using the Empirical Rule, we need to calculate how much of the given dataset falls within one standard deviation of the mean.
Given:
- Mean (\(\mu\)) = 10.38
- Standard deviation (\(\sigma\)) = 0.80
- One standard deviation range: \((\mu - \sigma, \mu + \sigma) = (10.38 - 0.80, 10.38 + 0.80) = (9.58, 11.18)\)
Now, let's examine the dataset and see how many data points fall within this range of (9.58, 11.18):
Dataset: 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
Count of data points in range (9.58, 11.18):
- 9.70: Within range
- 9.90: Within range
- 10.00: Within range
- 10.10: Within range
- 10.30: Within range
- 10.50: Within range
- 10.50: Within range
- 10.60: Within range
- 10.90: Within range
- 10.95: Within range
- 11.10: Within range
- 11.20: Within range
The following values fall within the range: 9.70, 9.90, 10.00, 10.10, 10.30, 10.50 (2 occurrences), 10.60, 10.90, 10.95, 11.10, 11.20.
Total count of data points that are within the range = 11
Total number of data points = 15
Percentage of data within one standard deviation of the mean: \[ \text{Percentage} = \left(\frac{11}{15}\right) \times 100 = 73.33% \]
According to the Empirical Rule, if the data is normally distributed, approximately 68% of the data should fall within one standard deviation of the mean.
Since 73.33% is close to 68%, we can conclude that:
A normal curve can be used to model the data, because this is approximately 68% of the data.
Therefore, the correct response is: "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."
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