at a certain factory, weekly wages (w) are normally distributed with a mean of $400 and a standard deviation…

at a certain factory, weekly wages (w) are normally distributed with a mean of $400 and a standard deviation of $50. find the probability that a worker selected at random makes between $400 and $550.\n99.7%\n95%\n68%\n250 300 350 400 450 500 550\np(400 < w < 550) = ?%\nbe sure to use the 68% - 95% - 99.7% rule and do not round.

at a certain factory, weekly wages (w) are normally distributed with a mean of $400 and a standard deviation of $50. find the probability that a worker selected at random makes between $400 and $550.\n99.7%\n95%\n68%\n250 300 350 400 450 500 550\np(400 < w < 550) = ?%\nbe sure to use the 68% - 95% - 99.7% rule and do not round.

Answer

Explanation:

Step1: Recall the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule for a normal distribution states that about 68% of the data lies within 1 standard - deviation of the mean, 95% within 2 standard - deviations, and 99.7% within 3 standard - deviations. The mean $\mu = 400$ and the standard deviation $\sigma=50$.

Step2: Calculate the number of standard - deviations

For $x = 550$, the z - score $z=\frac{x-\mu}{\sigma}=\frac{550 - 400}{50}=\frac{150}{50}=3$. The mean is $\mu = 400$ (z - score of 0).

Step3: Determine the probability

The area between the mean ($z = 0$) and $z = 3$ can be found using the 68 - 95 - 99.7 rule. The total area within 3 standard - deviations of the mean is 99.7%. The area on each side of the mean is $\frac{99.7%}{2}=49.85%$.

Answer:

49.85%