Question

consider the normal distribution curve.

which statements are true about the curve? check all that apply.

□ the standard deviation of the data is 64.
□ the variance of the data is 49.
□ the median is 64.
□ the data point 75 is less than one standard deviation from the mean.
□ the data point 50 is two standard deviations away from the mean.

Explanation:

1. Standard Deviation: The diagram shows \(\sigma = 7\), so the standard deviation is 7, not 64. This statement is false.
2. Variance: Variance is \(\sigma^{2}\). Given \(\sigma = 7\), variance \(= 7^{2}=49\). This statement is true.
3. Median in Normal Distribution: In a normal distribution, the mean, median, and mode are equal. The peak of the normal curve is at 64, so the median is 64. This statement is true.
4. Data Point 75: The mean is 64, standard deviation \(\sigma = 7\). The distance of 75 from the mean is \(75 - 64=11\). Since \(\sigma = 7\), \(11>7\), so 75 is more than one standard deviation from the mean. This statement is false.
5. Data Point 50: The distance of 50 from the mean (64) is \(64 - 50 = 14\). Since \(\sigma = 7\), \(\frac{14}{7}=2\), so 50 is two standard deviations away from the mean. This statement is true.

Answer:

• The variance of the data is 49.
• The median is 64.
• The data point 50 is two standard deviations away from the mean.