Problem # 2 (12.5 pts). For women aged 18-24, systolic blood
pressure (in mm Hg) are normally distributed with a mean of 114.8
and a standard deviation of 13.1 (based on data from the National
Health Survey). Hypertension is commonly defined as a systolic blood
pressure above 140. (Draw the bell curve).

a) If a sample of 40 women is selected in that age bracket is
randomly selected, find the probability that their mean systolic
blood pressure is between 139 and 140. (Draw the bell curve).

Let's solve the problem step by step.

### Given Data:
- Mean systolic blood pressure (\(\mu\)) = 114.8 mm Hg
- Standard deviation (\(\sigma\)) = 13.1 mm Hg
- Sample size (\(n\)) = 40
- Hypertension threshold = 140 mm Hg

We are asked to find the probability that the mean systolic blood pressure of a sample of 40 women is between 139 and 140 mm Hg.

### Step 1: Calculate the Standard Error of the Mean (SEM)
The standard error of the mean is given by:

\[
\text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{13.1}{\sqrt{40}} 
\]

### Step 2: Convert the Sample Mean to a Z-score
To find the probability, we'll convert the sample mean values (139 and 140) to Z-scores using the formula:

\[
Z = \frac{X - \mu}{\text{SEM}}
\]

where:
- \(X\) is the sample mean.

### Step 3: Find the Probability
After calculating the Z-scores, we'll use the standard normal distribution to find the probability that the sample mean lies between 139 and 140.

### Step 4: Draw the Bell Curve
We will also sketch the bell curve to visualize the problem.

Let's proceed with the calculations.### Results:
- **Z-score for 139 mm Hg:** \(Z_1 = 11.68\)
- **Z-score for 140 mm Hg:** \(Z_2 = 12.17\)
- **Probability that the mean systolic blood pressure is between 139 and 140 mm Hg:** \(0.0\)

### Interpretation:
The Z-scores of 11.68 and 12.17 are extremely high, which means these values are far in the tail of the normal distribution. The probability of the sample mean systolic blood pressure being between 139 and 140 mm Hg is practically zero. This indicates that it's extremely unlikely to get a sample mean in this range if the true population mean is 114.8 mm Hg.

### Bell Curve Visualization:
I will now draw a bell curve illustrating this distribution and the area under the curve between 139 and 140 mm Hg.Here is the bell curve representing the normal distribution of mean systolic blood pressure for women aged 18-24. The area between 139 and 140 mm Hg is highlighted in red, but as the probability calculation showed, this area is practically zero. This means that it's extremely unlikely for the mean systolic blood pressure of a sample of 40 women to fall within this range.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the probability change if the sample size were smaller?
2. What would the probability be for a different range, say between 135 and 137 mm Hg?
3. How does the Central Limit Theorem justify the use of the normal distribution here?
4. What is the effect of increasing the sample size on the standard error?
5. How can this information be used in medical research or public health?

**Tip:** Understanding Z-scores is crucial for interpreting probabilities in normal distributions. A higher absolute Z-score indicates a value further from the mean.

Calculate the probability that the mean systolic blood pressure of a sample of 40 women aged 18-24 falls between 139 and 140 mm Hg using normal distribution. Explore the use of SEM, Z-scores, and the Central Limit Theorem in solving this problem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation