Given:

• Mean ($\mu$) = 82
• Standard deviation ($\sigma$) = 5
• We are looking for $P(66 < x < 73)$, which is the probability that $x$ lies between 66 and 73.

Steps to solve:

1. Convert the values 66 and 73 to standard $z$-scores using the formula: $z = \frac{x - \mu}{\sigma}$ For $x = 66$: $z_1 = \frac{66 - 82}{5} = \frac{-16}{5} = -3.2$ For $x = 73$: $z_2 = \frac{73 - 82}{5} = \frac{-9}{5} = -1.8$

2. Now, use the standard normal distribution table (or a calculator) to find the probabilities corresponding to these $z$-scores:

   • $P(z_1 = -3.2) \approx 0.0007$
   • $P(z_2 = -1.8) \approx 0.0359$

3. The probability $P(66 < x < 73)$ is the difference between these probabilities: $P(66 < x < 73) = P(z_2) - P(z_1) = 0.0359 - 0.0007 = 0.0352$

So, $P(66 < x < 73) \approx 0.0352$, or about 3.52%.

Would you like more details on this? Here are 5 related questions:

1. How do you calculate probabilities for values outside the mean, such as $P(x > 85)$?
2. What is the significance of the standard normal distribution in statistics?
3. How do you interpret probabilities in a normal distribution?
4. How can z-scores be used to compare data from different normal distributions?
5. How can you calculate probabilities for non-standard normal distributions?

Tip: Always double-check z-score calculations and reference tables when determining probabilities for accurate results.