Recent study shows the diastolic blood pressure of women is normally distributed with a mean of 80.4 and a standard deviation of 9.5

Question

What proportions have bl lower that 71

Answer 1

To find the proportion of women with a diastolic blood pressure lower than 71, we can use the properties of the normal distribution.

Given:

• Mean (\(\mu\)) = 80.4
• Standard deviation (\(\sigma\)) = 9.5
• Value to compare (\(X\)) = 71

First, we need to calculate the z-score, which is given by the formula:

\[ z = \frac{X - \mu}{\sigma} \]

Substituting in the values:

\[ z = \frac{71 - 80.4}{9.5} = \frac{-9.4}{9.5} \approx -0.9895 \]

Next, we will use the z-score to find the proportion of the population that falls below this z-score. We can look up the z-score in a standard normal distribution table or use a calculator to find the cumulative probability.

Using a z-table or calculator, a z-score of approximately -0.99 corresponds to a cumulative probability of about 0.1611.

This means that approximately 16.11% of women have a diastolic blood pressure lower than 71.

Answer 2

What proportion have BP between 69 and 91

Answer 3

To find the proportion of women with a diastolic blood pressure between 69 and 91, we will calculate the z-scores for both values (69 and 91) and then find the cumulative probabilities associated with these z-scores.

Step 1: Calculate the z-scores

1. For \(X = 69\):

\[ z = \frac{X - \mu}{\sigma} = \frac{69 - 80.4}{9.5} = \frac{-11.4}{9.5} \approx -1.2 \]

2. For \(X = 91\):

\[ z = \frac{X - \mu}{\sigma} = \frac{91 - 80.4}{9.5} = \frac{10.6}{9.5} \approx 1.1158 \]

Step 2: Find cumulative probabilities

Using a z-table or calculator:

1. Cumulative probability for \(z \approx -1.2\) is approximately \(0.1151\).

2. Cumulative probability for \(z \approx 1.12\) is approximately \(0.8686\).

Step 3: Calculate the proportion between the two z-scores

To find the proportion of women with a diastolic blood pressure between 69 and 91, subtract the cumulative probability for \(z = -1.2\) from the cumulative probability for \(z = 1.12\):

\[ P(69 < BP < 91) = P(Z < 1.12) - P(Z < -1.2) \approx 0.8686 - 0.1151 = 0.7535 \]

Conclusion

Approximately 75.35% of women have a diastolic blood pressure between 69 and 91.

Answer 4

Greater than 90 is classified as hypertension. What proportion of woken have hypertension

Answer 5

To find the proportion of women with a diastolic blood pressure greater than 90 (classified as hypertension), we will calculate the z-score for 90 and then find the corresponding cumulative probability.

Step 1: Calculate the z-score

For \(X = 90\):

\[ z = \frac{X - \mu}{\sigma} = \frac{90 - 80.4}{9.5} = \frac{9.6}{9.5} \approx 1.0105 \]

Step 2: Find cumulative probability

Using a z-table or calculator:

The cumulative probability for \(z \approx 1.01\) is approximately \(0.8438\). This value represents the proportion of women with a diastolic blood pressure less than or equal to 90.

Step 3: Calculate the proportion greater than 90

To find the proportion of women with a diastolic blood pressure greater than 90:

\[ P(BP > 90) = 1 - P(Z < 1.01) \approx 1 - 0.8438 = 0.1562 \]

Conclusion

Approximately 15.62% of women have hypertension (diastolic blood pressure greater than 90).

Answer 6

Is it unusual for a woman to have a BP lower than 67? Why

Answer 7

To determine whether it is unusual for a woman to have a diastolic blood pressure lower than 67, we will calculate the z-score for 67 and analyze its meaning in the context of the normal distribution.

Step 1: Calculate the z-score

Given:

• Mean (\(\mu\)) = 80.4
• Standard deviation (\(\sigma\)) = 9.5
• Value (\(X\)) = 67

Using the z-score formula:

\[ z = \frac{X - \mu}{\sigma} = \frac{67 - 80.4}{9.5} = \frac{-13.4}{9.5} \approx -1.4158 \]

Step 2: Find the cumulative probability

Using a z-table or calculator, the cumulative probability for \(z \approx -1.42\) is approximately \(0.0772\). This means that about 7.72% of women have a diastolic blood pressure lower than 67.

Step 3: Interpret the result

In general, a probability of less than 5% is often considered unusual or rare in statistical terms. Since 7.72% is slightly above this threshold, it can be argued that while it is relatively uncommon for a woman to have a diastolic blood pressure lower than 67, it is not extremely unusual.

Conclusion

So, while it is on the lower end of the spectrum and not very common, it is not considered "unusual" in the strict statistical sense since the proportion (about 7.72%) is higher than the typical cutoff of 5% for defining unusual events. The cutoff can vary depending on the context and the specific definitions used, but statistically, values that fall further than 2 standard deviations from the mean are often referred to as unusual. Since \(-1.42\) is just slightly more than 1.4 standard deviations away, it does not strongly indicate an unusual occurrence.