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

Explanation:

Step1: Calculate the number of standard deviations

The mean $\mu = 400$, standard deviation $\sigma=50$. For $x = 550$, $z=\frac{x-\mu}{\sigma}=\frac{550 - 400}{50}=3$.

Step2: Use the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that for a normal distribution:

• Approximately 68% of the data is within 1 standard deviation of the mean ($\mu\pm\sigma$)
• Approximately 95% of the data is within 2 standard deviations of the mean ($\mu\pm2\sigma$)
• Approximately 99.7% of the data is within 3 standard deviations of the mean ($\mu\pm3\sigma$)

The probability that a value is within $3$ standard deviations of the mean is 99.7%. Since the normal distribution is symmetric about the mean, the probability that a value is between the mean ($\mu = 400$) and $3$ standard deviations above the mean ($x = 550$) is $\frac{99.7%}{2}=49.85%$

Answer:

$49.85$