A random variable $\mathrm{x}$ is normally distributed with $\mu=240$ and $\sigma=25$. Find the indicated probabilities:
1. $P(x<229)$
2. $P(x>267)$
3. $P(231
Solution
Step 1 ：First, we need to standardize the random variable $x$ by subtracting the mean $\mu$ and dividing by the standard deviation $\sigma$. This gives us the z-score, which follows a standard normal distribution with mean 0 and standard deviation 1.
Step 2 ：The z-score is calculated as $z = \frac{x - \mu}{\sigma}$.
Step 3 ：For $P(x<229)$, we calculate the z-score as $z = \frac{229 - 240}{25} = -0.44$.
Step 4 ：We look up this z-score in the standard normal distribution table, or use a calculator or software that can calculate it, to find the probability. The probability $P(x<229)$ is equal to $P(Z<-0.44)$, which is approximately 0.33.
Step 5 ：For $P(x>267)$, we calculate the z-score as $z = \frac{267 - 240}{25} = 1.08$.
Step 6 ：The probability $P(x>267)$ is equal to $1 - P(Z<1.08)$, because the total probability under the standard normal curve is 1. Looking up $P(Z<1.08)$ in the standard normal distribution table or using a calculator or software, we find it is approximately 0.86. So, $P(x>267)$ is approximately $1 - 0.86 = 0.14$.
Step 7 ：For $P(231
Step 8 ：The probability $P(231
Step 9 ：So, the final answers are $P(x<229) \approx 0.33$, $P(x>267) \approx 0.14$, and $P(231
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Solution
Step 1 ：First, we need to standardize the random variable $x$ by subtracting the mean $\mu$ and dividing by the standard deviation $\sigma$. This gives us the z-score, which follows a standard normal distribution with mean 0 and standard deviation 1.
Step 2 ：The z-score is calculated as $z = \frac{x - \mu}{\sigma}$.
Step 3 ：For $P(x<229)$, we calculate the z-score as $z = \frac{229 - 240}{25} = -0.44$.
Step 4 ：We look up this z-score in the standard normal distribution table, or use a calculator or software that can calculate it, to find the probability. The probability $P(x<229)$ is equal to $P(Z<-0.44)$, which is approximately 0.33.
Step 5 ：For $P(x>267)$, we calculate the z-score as $z = \frac{267 - 240}{25} = 1.08$.
Step 6 ：The probability $P(x>267)$ is equal to $1 - P(Z<1.08)$, because the total probability under the standard normal curve is 1. Looking up $P(Z<1.08)$ in the standard normal distribution table or using a calculator or software, we find it is approximately 0.86. So, $P(x>267)$ is approximately $1 - 0.86 = 0.14$.
Step 7 ：For $P(231 Step 8 ：The probability $P(231 Step 9 ：So, the final answers are $P(x<229) \approx 0.33$, $P(x>267) \approx 0.14$, and $P(231
