consider the normal distribution curve. which statements are true about the curve? check all that apply. the…

Explanation:

Step1: Recall standard - deviation and variance relationship

The variance $\sigma^{2}$ is related to the standard deviation $\sigma$ by the formula $\sigma^{2}=\sigma\times\sigma$. Given $\sigma = 7$, then $\sigma^{2}=7\times7 = 49$.

Step2: Recall properties of normal distribution

In a normal distribution, the mean, median, and mode are equal. The peak of the normal - distribution curve is at the mean/median/mode. Here, the peak is at $x = 64$, so the median is 64.

Step3: Calculate number of standard deviations from the mean

The mean is 64 and the standard deviation $\sigma=7$. For a data - point $x$, the number of standard deviations from the mean is $z=\frac{x - \mu}{\sigma}$, where $\mu$ is the mean. For $x = 75$, $z=\frac{75 - 64}{7}=\frac{11}{7}\approx1.57>1$. For $x = 50$, $z=\frac{50 - 64}{7}=\frac{- 14}{7}=-2$.

Answer:

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