Final answer:
The probabilities of a woman having diastolic blood pressure less than 60 mm Hg, greater than 90 mm Hg, and between 60 and 90 mm Hg can be found by calculating the respective Z-scores and using the standard normal distribution table. For the 90% bounds, the Z-scores corresponding to the 5th and 95th percentiles are used.
Step-by-step explanation:
To answer the student's question about the probability of a woman having a certain diastolic blood pressure within a specified range, we utilize the properties of the normal distribution. Given that the diastolic blood pressure among females in the United States is normally distributed with a mean (μ) of 77 mm Hg and a standard deviation (σ) of 11.6 mm Hg, we can calculate the requested probabilities.
(a) Probability of blood pressure < 60 mm Hg
First, we calculate the Z-score for a blood pressure of 60 mm Hg:
Z = (X - μ) / σ
Z = (60 - 77) / 11.6
Z = -1.47
Using the standard normal distribution table or a calculator, we find P(Z < -1.47). This gives us the probability that a randomly selected woman has a diastolic blood pressure less than 60 mm Hg.
(b) Probability of blood pressure > 90 mm Hg
Next, we calculate the Z-score for a blood pressure of 90 mm Hg:
Z = (90 - 77) / 11.6
Z = 1.12
We find P(Z > 1.12) to obtain the probability of a woman having a diastolic blood pressure greater than 90 mm Hg.
(c) Probability of blood pressure between 60 and 90 mm Hg
To find the probability of a blood pressure between 60 and 90 mm Hg, we use the previously calculated Z-scores and find P(-1.47 < Z < 1.12). This is achieved by subtracting P(Z < -1.47) from P(Z < 1.12).
(d) Bounds for 90% of the distribution
To find the bounds within which 90% of women's diastolic blood pressures lie, we look for the Z-scores that correspond to the 5th and 95th percentiles (since 90% is in the middle, leaving 5% at each tail). We use a Z-table or calculator to find these Z-scores and convert them back to diastolic blood pressure values using the formula:
X = (Z * σ) + μ