Given a normal distribution with a mean of 40 and a standard deviation of 6, what is the range that contains approximately 95% of the data?

Rationale

The range is from 28 to 52.

In a normal distribution, approximately 95% of the data falls within two standard deviations of the mean. With a mean of 40 and a standard deviation of 6, we calculate the range by subtracting and adding 12 (2 times the standard deviation) from the mean, resulting in a range from 28 to 52.

A (The range is from 28 to 52.) CORRECT

This option correctly represents the range that includes approximately 95% of the data in a normal distribution. Calculating 40 - 12 gives 28, and 40 + 12 gives 52, confirming that this range accurately encompasses the significant majority of the distribution.

B (The range is from 34 to 46.) YOUR ANSWER

This range only accounts for one standard deviation from the mean (40 ± 6), which would only cover about 68% of the data, not the 95% that is required. Therefore, it does not capture the wider spread necessary to reflect the majority of the distribution.

C (The range is from 22 to 58.) YOUR ANSWER

While this range does extend beyond two standard deviations, it incorrectly includes data that falls outside the scope of the 95% confidence interval. The calculated extremes should be 28 and 52, making this option too broad to accurately represent the desired data range.

D (The range is from 30 to 50.) YOUR ANSWER

Similar to option B, this range does not extend far enough to cover two standard deviations from the mean. It only includes a portion of the data, thus failing to meet the requirement of encompassing approximately 95% of the distribution.

Conclusion

In summary, the range from 28 to 52 correctly reflects the spread of data within two standard deviations of the mean in a normal distribution with a mean of 40 and a standard deviation of 6. Understanding this concept is crucial for accurately interpreting data distributions and their implications in statistical analysis.

Given a normal distribution with a mean of 40 and a standard deviation of 6, what is the range that contains approximately 95% of the data?

The range is from 28 to 52.

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