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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.
$P(35<X<57)=◻$
B.
c.
(Round to four decimal places as needed.)
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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 $\boxed{{\displaystyle 0.8253}}$.