The systolic blood pressures of the patients at a hospital are normally distributed with a mean of

136 mm Hg and a standard deviation of 13.4 mm Hg. Find the two blood pressures having these
properties: the mean is midway between them and 90% of all blood pressures are between them.

Looking up under the table of "areas under the normal distribution" for ±.05 in the smaller area, 90% of the scores in a normal distribution fall within the mean ± 1.645 SD.

You should be able to do the calculations.

To find the two blood pressures having the given properties, we can use the normal distribution and z-scores. Here are the steps to solve the problem:

Step 1: Identify the z-scores corresponding to the desired percentiles.
We are given that 90% of all blood pressures are between the two pressures, which means 5% will be below the lower pressure and 5% will be above the higher pressure.

To find the z-scores corresponding to these percentiles, we can use a standard normal distribution table or calculator. The z-score for the lower pressure will be the z-score at the 5th percentile (0.05) and the z-score for the higher pressure will be the z-score at the 95th percentile (0.95).

Step 2: Calculate the actual values using z-scores.
Once we have the z-scores, we can use the formula for z-scores to calculate the actual values (blood pressures).

The formula for converting a z-score to an actual value is:

x = μ + (z * σ)

Where:
x represents the actual value (blood pressure),
μ is the mean (136 mm Hg),
z is the z-score,
σ is the standard deviation (13.4 mm Hg).

Step 3: Calculate the mean midway between the two pressures.
Given that the mean is midway between the two pressures, we can calculate it by taking the average of the two pressure values obtained in Step 2.

Let's calculate the two blood pressures with the properties mentioned:

Step 1: Finding the z-scores:
The z-score for the lower pressure can be found using a standard normal distribution table or calculator. For a percentile of 0.05, the z-score is approximately -1.645.
The z-score for the higher pressure can be found similarly. For a percentile of 0.95, the z-score is approximately 1.645.

Step 2: Calculating the actual values:
Using the formula above, we can calculate the actual values:

x_lower = μ + (z_lower * σ)
= 136 + (-1.645 * 13.4)
≈ 113.63 mm Hg

x_higher = μ + (z_higher * σ)
= 136 + (1.645 * 13.4)
≈ 158.37 mm Hg

So, the lower blood pressure is approximately 113.63 mm Hg, and the higher blood pressure is approximately 158.37 mm Hg.

Step 3: Calculating the mean midway between the two pressures:
The mean is midway between the two pressures. Therefore, the average of the lower and higher pressures is:

mean = (x_lower + x_higher) / 2
= (113.63 + 158.37) / 2
≈ 136 mm Hg.

Therefore, the mean is approximately 136 mm Hg.

To summarize, the two blood pressures meeting the given conditions are approximately 113.63 mm Hg and 158.37 mm Hg, with the mean being approximately 136 mm Hg.