analyzing a normal distribution curve\nconsider the normal distribution curve.\nwhich statements are true…

analyzing a normal distribution curve\nconsider the normal distribution curve.\nwhich statements are true about the curve? check all that apply.\nσ = 7\nthe standard deviation of the data is 64.\nthe variance of the data is 49.\nthe median is 64.\nthe data point 75 is less than one standard deviation from the mean.\nthe data point 50 is two standard deviations away from the mean.\n50 57 64 71 78

analyzing a normal distribution curve\nconsider the normal distribution curve.\nwhich statements are true about the curve? check all that apply.\nσ = 7\nthe standard deviation of the data is 64.\nthe variance of the data is 49.\nthe median is 64.\nthe data point 75 is less than one standard deviation from the mean.\nthe data point 50 is two standard deviations away from the mean.\n50 57 64 71 78

Answer

Explanation:

Step1: Recall standard - deviation and variance relationship

The variance $\sigma^{2}$ is related to the standard deviation $\sigma$ by $\sigma^{2}=\sigma\times\sigma$. 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 64, so the median is 64.

Step3: Calculate standard - deviation distances

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.