Question

The following dataset has a population standard deviation of 6.47 and a mean of 43.33. What percentage of the data lies within 2 standard deviations of the mean? Using the Empirical Rule, determine if a normal curve can be used to model the data. 43.16 46.10 43.01 35.65 53.15 43.19 47.19 53.75 43.57 35.60 37.28 44.47 44.64 38.51 32.34 47.81 31.04 50.02 49.06 47.25 (1 point) Responses 80 percent of the data falls within 2 standard deviations of the mean. A normal curve can not be used to model the data, because this is not approximately 95% of the data. 80 percent of the data falls within 2 standard deviations of the mean. A normal curve can not be used to model the data, because this is not approximately 95% of the data. 43.34 percent of the data falls within 2 standard deviations of the mean. A normal curve can not be used to model the data, because this is not approximately 95% of the data. 43.34 percent of the data falls within 2 standard deviations of the mean. A normal curve can not be used to model the data, because this is not approximately 95% of the data. 65 percent of the data falls within 2 standard deviations of the mean. A normal curve can not be used to model the data, because this is not approximately 95% of the data. 65 percent of the data falls within 2 standard deviations of the mean. A normal curve can not be used to model the data, because this is not approximately 95% of the data. 100 percent of the data falls within 2 standard deviations of the mean. A normal curve can be used to model the data, because this is approximately 95% of the data. 100 percent of the data falls within 2 standard deviations of the mean. A normal curve can be used to model the data, because this is approximately 95% of the data. Skip to navigation

Answer 1

To determine the percentage of the data that falls within 2 standard deviations of the mean using the Empirical Rule (also known as the 68-95-99.7 rule), we first need to understand the rule itself:

• Approximately 68% of the data lies within 1 standard deviation of the mean.
• Approximately 95% of the data lies within 2 standard deviations of the mean.
• Approximately 99.7% of the data lies within 3 standard deviations of the mean.

Given:

• Mean (\( \mu \)) = 43.33
• Population Standard Deviation (\( \sigma \)) = 6.47

To find the range within 2 standard deviations:

• Lower Bound: \( \mu - 2\sigma = 43.33 - 2(6.47) = 43.33 - 12.94 = 30.39 \)
• Upper Bound: \( \mu + 2\sigma = 43.33 + 2(6.47) = 43.33 + 12.94 = 56.27 \)

Next, we can check how many of the provided data points fall within the range of 30.39 to 56.27.

The dataset provided is:

• 43.16, 46.10, 43.01, 35.65, 53.15, 43.19, 47.19, 53.75, 43.57,
• 35.60, 37.28, 44.47, 44.64, 38.51, 32.34, 47.81, 31.04,
• 50.02, 49.06, 47.25

Now we can count how many of these values fall within the bounds of 30.39 and 56.27:

1. 43.16 (in)
2. 46.10 (in)
3. 43.01 (in)
4. 35.65 (in)
5. 53.15 (in)
6. 43.19 (in)
7. 47.19 (in)
8. 53.75 (in)
9. 43.57 (in)
10. 35.60 (in)
11. 37.28 (in)
12. 44.47 (in)
13. 44.64 (in)
14. 38.51 (in)
15. 32.34 (in)
16. 47.81 (in)
17. 31.04 (in)
18. 50.02 (in)
19. 49.06 (in)
20. 47.25 (in)

All 20 data points fall within the range of 30.39 to 56.27.

Thus, 100% of the data lies within 2 standard deviations of the mean.

Now, we evaluate if a normal curve can be used to model the data. The Empirical Rule states that for a normal distribution, approximately 95% of the data should lie within 2 standard deviations of the mean. Since we found that 100% of the data falls within this range, we can say that a normal curve can effectively model the data as it is consistent with the expectations of the Empirical Rule.

The correct response would therefore be:

100 percent of the data falls within 2 standard deviations of the mean. A normal curve can be used to model the data, because this is approximately 95% of the data.