A data set is normally distributed with a mean of 30 and a standard deviation of 3. Which percentage of the data falls between the range of 24 and 30?

Rationale

48 percent of the data falls between the range of 24 and 30.

In a normal distribution, the area under the curve represents the percentage of data points. The range between one standard deviation below the mean (24) and the mean (30) encompasses a specific portion of the distribution, which corresponds to the empirical rule.

A (48) CORRECT

This choice represents the percentage of data that falls between the mean and one standard deviation below the mean in a normal distribution. Since 34% of the data lies between the mean (30) and one standard deviation above the mean (33), and another 14% lies between the mean and one standard deviation below (24), the total percentage from 24 to 30 is 34% + 14% = 48%.

B (34) YOUR ANSWER

This choice corresponds to the percentage of data that falls between the mean (30) and one standard deviation above (33). While this is a significant portion of the distribution, it does not account for the data between the mean and one standard deviation below (24), which contributes additional percentage to the range.

C (25) YOUR ANSWER

This figure does not align with the standard percentages defined by the empirical rule for a normal distribution. It underestimates the data range from 24 to 30, failing to capture the full area represented between the mean and one standard deviation below.

D (68) YOUR ANSWER

This choice indicates the percentage of data that falls within one standard deviation of the mean, covering both sides of the mean from 27 to 33. It is not relevant for the specific range from 24 to 30, which is only one half of the standard deviation range.

Conclusion

In a normal distribution, the percentage of data that falls between the mean and one standard deviation below is crucial for understanding the distribution of values. The correct percentage of 48% reflects the combined area of 34% above the mean and 14% below it, clearly demonstrating the application of the empirical rule in solving for specific data ranges within a normally distributed data set.

A data set is normally distributed with a mean of 30 and a standard deviation of 3. Which percentage of the data falls between the range of 24 and 30?

48 percent of the data falls between the range of 24 and 30.

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