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Assume the random variable $X$ is normally distributed with mean $\mu=50$ and standard deviation $\sigma=7$. Compute the probability. Be sure to draw a normal curve with the area corresponding to the probability shaded.
\[
\mathrm{P}(35<\mathrm{X}<57)
\]
Click the icon to view a table of areas under the normal curve.
Which of the following normal curves corresponds to $\mathrm{P}(35<\mathrm{X}<57)$ ?
A.
$P(35
Solution
Step 1 ：Given that the random variable $X$ is normally distributed with mean $\mu=50$ and standard deviation $\sigma=7$, we are asked to compute the probability $P(35
Step 2 ：We first standardize the values 35 and 57 using the formula $Z = \frac{X - \mu}{\sigma}$.
Step 3 ：For $X=35$, we get $Z_1 = \frac{35 - 50}{7} = -2.142857142857143$.
Step 4 ：For $X=57$, we get $Z_2 = \frac{57 - 50}{7} = 1.0$.
Step 5 ：We then look up the probabilities corresponding to these Z-scores in the standard normal distribution table.
Step 6 ：For $Z_1 = -2.142857142857143$, we get $P_1 = 0.016062285603828316$.
Step 7 ：For $Z_2 = 1.0$, we get $P_2 = 0.8413447460685429$.
Step 8 ：We subtract these probabilities to find the probability of $X$ falling between 35 and 57, i.e., $P = P_2 - P_1 = 0.8413447460685429 - 0.016062285603828316 = 0.8252824604647147$.
Step 9 ：Thus, the probability that the random variable $X$ falls between 35 and 57 is approximately \(\boxed{0.8253}\).
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Solution
Step 1 ：Given that the random variable $X$ is normally distributed with mean $\mu=50$ and standard deviation $\sigma=7$, we are asked to compute the probability $P(35 Step 2 ：We first standardize the values 35 and 57 using the formula $Z = \frac{X - \mu}{\sigma}$. Step 3 ：For $X=35$, we get $Z_1 = \frac{35 - 50}{7} = -2.142857142857143$. Step 4 ：For $X=57$, we get $Z_2 = \frac{57 - 50}{7} = 1.0$. Step 5 ：We then look up the probabilities corresponding to these Z-scores in the standard normal distribution table. Step 6 ：For $Z_1 = -2.142857142857143$, we get $P_1 = 0.016062285603828316$. Step 7 ：For $Z_2 = 1.0$, we get $P_2 = 0.8413447460685429$. Step 8 ：We subtract these probabilities to find the probability of $X$ falling between 35 and 57, i.e., $P = P_2 - P_1 = 0.8413447460685429 - 0.016062285603828316 = 0.8252824604647147$. Step 9 ：Thus, the probability that the random variable $X$ falls between 35 and 57 is approximately \(\boxed{0.8253}\).
