Questions: Given a normal distribution with μ=45 and σ=4, complete parts (a) through (d). a. What is the probability that X>38 ? P(X>38)= (Round to four decimal places as needed.) b. What is the probability that X<37 ? P(X<37)= (Round to four decimal places as needed.) c. For this distribution, 7% of the values are less than what X-value? X= (Round to the nearest integer as needed.) d. Between what two X-values (symmetrically distributed around the mean) are 70% of the values? For this distribution, 70% of the values are between X= and X= .

Given a normal distribution with μ=45 and σ=4, complete parts (a) through (d).
a. What is the probability that X>38 ?
P(X>38)= (Round to four decimal places as needed.)
b. What is the probability that X<37 ?
P(X<37)= (Round to four decimal places as needed.)
c. For this distribution, 7% of the values are less than what X-value?
X= (Round to the nearest integer as needed.)
d. Between what two X-values (symmetrically distributed around the mean) are 70% of the values?
For this distribution, 70% of the values are between X= and X= .

Transcript text: Given a normal distribution with $\mu=45$ and $\sigma=4$, complete parts (a) through (d). a. What is the probability that $X>38$ ? $P(X>38)=$ $\square$ (Round to four decimal places as needed.) b. What is the probability that $X<37$ ? $P(X<37)=$ $\square$ (Round to four decimal places as needed.) c. For this distribution, $7 \%$ of the values are less than what $X$-value? $X=\square$ $\square$ (Round to the nearest integer as needed.) d. Between what two $X$-values (symmetrically distributed around the mean) are $70 \%$ of the values? For this distribution, $70 \%$ of the values are between $X=$ and $X=$ $\square$ .

Solution

Solution Steps

Step 1: Calculate $ P(X > 38) $

To find the probability that $ X > 38 $ for a normal distribution with mean $ \mu = 45 $ and standard deviation $ \sigma = 4 $, we first calculate the Z-score for $ X = 38 $:

\[ Z = \frac{X - \mu}{\sigma} = \frac{38 - 45}{4} = -1.75 \]

Using the cumulative distribution function (CDF) for the standard normal distribution, we find:

\[ P(X > 38) = 1 - \Phi(-1.75) = 1 - 0.0401 = 0.9599 \]

Step 2: Calculate $ P(X < 37) $

To find the probability that $ X < 37 $, we calculate the Z-score for $ X = 37 $:

\[ Z = \frac{X - \mu}{\sigma} = \frac{37 - 45}{4} = -2.0 \]

Using the CDF for the standard normal distribution, we find:

\[ P(X < 37) = \Phi(-2.0) = 0.0228 \]

Step 3: Find $ X $ for which 7% of the values are less than $ X $

We need to find the $ X $-value such that 7% of the values are less than $ X $. This corresponds to finding the 7th percentile of the distribution. The Z-score for the 7th percentile is approximately $ -1.4758 $.

Using the Z-score formula, we solve for $ X $:

\[ X = \mu + Z \cdot \sigma = 45 + (-1.4758) \cdot 4 = 39.0968 \]

Rounding to the nearest integer, we find:

\[ X = 39 \]