Question

the graph of a normally distributed random variable is given. use the graph to answer the questions that follow. what is the mean? what is the standard deviation? what percentage of values should fall between 56 and 62? which value has a z - score of - 2?

Explanation:

Step1: Identify mean of normal distribution

In a normal - distribution graph, the mean is at the center of the symmetric curve. Here, the center value is 56. So the mean $\mu = 56$.

Step2: Calculate standard deviation

The distance between consecutive marked values is the standard deviation. From 56 to 62 (or from 50 to 56), the difference is 6. So the standard deviation $\sigma=6$.

Step3: Find percentage between 56 and 62

The empirical rule for a normal distribution states that about 68% of the data lies within 1 standard - deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard - deviations ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard - deviations ($\mu\pm3\sigma$). The interval from 56 to 62 is from the mean ($\mu = 56$) to $\mu+\sigma$ (since $\mu = 56$ and $\sigma = 6$). The percentage of data from $\mu$ to $\mu+\sigma$ is $\frac{68\%}{2}=34\%$.

Step4: Find value with z - score of - 2

The z - score formula is $z=\frac{x-\mu}{\sigma}$. Given $z=-2$, $\mu = 56$, and $\sigma = 6$. Rearranging the formula for $x$ gives $x=\mu+z\sigma$. Substituting the values, we get $x = 56+( - 2)\times6=56 - 12 = 44$.

Answer:

Mean: 56
Standard deviation: 6
Percentage between 56 and 62: 34
Value with z - score of - 2: 44