Question

analyzing a
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:

Step1: Recall the relationship between variance and standard deviation

Variance \(\sigma^{2}\), given \(\sigma = 7\), then \(\sigma^{2}=7^{2}=49\).

Step2: Properties of normal distribution

In a normal distribution, mean = median. The peak of the normal - distribution curve is at \(x = 64\), so mean=\(64\), median=\(64\).

Step3: Calculate the number of standard deviations from the mean

For a data point \(x\), the number of standard deviations from the mean \(z=\frac{x-\mu}{\sigma}\).

• For \(x = 75\), \(\mu = 64\), \(\sigma=7\), \(z=\frac{75 - 64}{7}=\frac{11}{7}\approx1.57>1\).
• For \(x = 50\), \(z=\frac{50 - 64}{7}=\frac{- 14}{7}=-2\) (the absolute value \(|z| = 2\))

Answer:

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