a normal distribution has a mean of 4 and a standard deviation of 1. what percent of values are from 4 to 7…

Answer:

49.87

Explanation:

Step1: Calculate z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu$ is the mean, $\sigma$ is the standard deviation, and $x$ is the value from the data set. For $x = 4$, $z_1=\frac{4 - 4}{1}=0$. For $x = 7$, $z_2=\frac{7 - 4}{1}=3$.

Step2: Use the standard normal table

The standard - normal table gives the cumulative probability $P(Z\leq z)$. $P(Z\leq0)=0.5$ (by the properties of the standard normal distribution, the mean of the standard normal distribution $Z$ has a cumulative probability of 0.5). $P(Z\leq3)\approx0.9987$ from the standard - normal table.

Step3: Find the probability between the two z - scores

$P(0\leq Z\leq3)=P(Z\leq3)-P(Z\leq0)$. $P(0\leq Z\leq3)=0.9987 - 0.5=0.4987$. To convert this probability to a percentage, we multiply by 100. So the percentage of values from 4 to 7 is $0.4987\times100 = 49.87%$.