Question 14 of 17 (1 point) I Question Attempt: 1 of Unlimited
Systolic Blood Pressure Assume that the mean systolic blood pressure of normal adults is 120 millimeters of mercury (mmHg) and the standard deviation is 5.6. Assume the variable is normally distributed. Round the answers to at least 4 decimal places and intermediate $z$-value calculations to 2 decimal places.
Part 1 of 3
If an individual is selected, find the probability that the individual's pressure will be between 118.2 and $121.2 \mathrm{mmHg}$.
\[
P(118.2< X< 121.2)=0.2109
\]

Part: $1 / 3$

Part 2 of 3
If a sample of 38 adults is randomly selected, find the probability that the sample mean will be between 118.2 and $121.2 \mathrm{mmHg}$. Assume that the sample is taken from a large population and the correction factor can be ignored.
\[
P(118.2< \bar{X}< 121.2)=\square
\]
Save For Later
Submit Ass

Step 1 ：Given that the mean systolic blood pressure of normal adults is \( \mu = 120 \) millimeters of mercury (mmHg) and the standard deviation is \( \sigma = 5.6 \).

Step 2 ：A sample of 38 adults is randomly selected, denoted as \( n = 38 \).

Step 3 ：We are asked to find the probability that the sample mean will be between 118.2 and 121.2 mmHg, denoted as \( X1 = 118.2 \) and \( X2 = 121.2 \).

Step 4 ：First, calculate the z-scores for \( X1 \) and \( X2 \) using the formula \( z = \frac{X - \mu}{\sigma / \sqrt{n}} \).

Step 5 ：For \( X1 = 118.2 \), the z-score \( z1 = \frac{X1 - \mu}{\sigma / \sqrt{n}} = -1.9814187866685964 \).

Step 6 ：For \( X2 = 121.2 \), the z-score \( z2 = \frac{X2 - \mu}{\sigma / \sqrt{n}} = 1.3209458577790696 \).

Step 7 ：Next, calculate the cumulative distribution function (CDF) for \( z1 \) and \( z2 \), denoted as \( P1 \) and \( P2 \) respectively.

Step 8 ：For \( z1 = -1.9814187866685964 \), the CDF \( P1 = 0.02377216433392225 \).

Step 9 ：For \( z2 = 1.3209458577790696 \), the CDF \( P2 = 0.906740291152752 \).

Step 10 ：Finally, calculate the probability that the sample mean will be between 118.2 and 121.2 by subtracting \( P1 \) from \( P2 \), denoted as \( P = P2 - P1 = 0.8829681268188297 \).

Step 11 ：The probability that the sample mean of systolic blood pressure of 38 adults will be between 118.2 and 121.2 mmHg is approximately \(\boxed{0.8829}\).