Answer:
Explained below.
Step-by-step explanation:
Let <em>X</em> = systolic blood pressure measurements.
It is provided that,
.
(a)
Compute the percentage of measurements that are between 71 and 89 as follows:
![P(71](https://tex.z-dn.net/?f=P%2871%3CX%3C89%29%3DP%28%5Cfrac%7B71-80%7D%7B3%7D%3C%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B89-80%7D%7B3%7D%29)
![=P(-3](https://tex.z-dn.net/?f=%3DP%28-3%3CZ%3C3%29%5C%5C%3DP%28Z%3C3%29-P%28Z%3C-3%29%5C%5C%3D0.99865-0.00135%5C%5C%3D0.9973)
The percentage is, 0.9973 × 100 = 99.73%.
Thus, the percentage of measurements that are between 71 and 89 is 99.73%.
(b)
Compute the probability that a person's blood systolic pressure measures more than 89 as follows:
![P(X>89)=P(\frac{X-\mu}{\sigma}>\frac{89-80}{3})](https://tex.z-dn.net/?f=P%28X%3E89%29%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3E%5Cfrac%7B89-80%7D%7B3%7D%29)
![=P(Z>3)\\=1-P(Z](https://tex.z-dn.net/?f=%3DP%28Z%3E3%29%5C%5C%3D1-P%28Z%3C3%29%5C%5C%3D1-0.99865%5C%5C%3D0.00135%5C%5C%5Capprox%200.0014)
Thus, the probability that a person's blood systolic pressure measures more than 89 is 0.0014.
(c)
Compute the probability that a person's blood systolic pressure being at most 75 as follows:
Apply continuity correction:
![P(X\leq 75)=P(X](https://tex.z-dn.net/?f=P%28X%5Cleq%2075%29%3DP%28X%3C75-0.5%29)
![=P(X](https://tex.z-dn.net/?f=%3DP%28X%3C74.5%29%5C%5C%5C%5C%3DP%28%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D%3C%5Cfrac%7B74.5-80%7D%7B3%7D%29%5C%5C%5C%5C%3DP%28Z%3C-1.83%29%5C%5C%5C%5C%3D0.03362%5C%5C%5C%5C%5Capprox%200.034)
Thus, the probability that a person's blood systolic pressure being at most 75 is 0.034.
(d)
Let <em>x</em> be the blood pressure required.
Then,
P (X < x) = 0.15
⇒ P (Z < z) = 0.15
⇒ <em>z</em> = -1.04
Compute the value of <em>x</em> as follows:
![z=\frac{x-\mu}{\sigma}\\\\-1.04=\frac{x-80}{3}\\\\x=80-(1.04\times3)\\\\x=76.88\\\\x\approx 76.9](https://tex.z-dn.net/?f=z%3D%5Cfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D%5C%5C%5C%5C-1.04%3D%5Cfrac%7Bx-80%7D%7B3%7D%5C%5C%5C%5Cx%3D80-%281.04%5Ctimes3%29%5C%5C%5C%5Cx%3D76.88%5C%5C%5C%5Cx%5Capprox%2076.9)
Thus, the 15% of patients are expected to have a blood pressure below 76.9.
(e)
A <em>z</em>-score more than 2 or less than -2 are considered as unusual.
Compute the <em>z</em> score for
as follows:
![z=\frac{\bar x-\mu}{\sigma/\sqrt{n}}](https://tex.z-dn.net/?f=z%3D%5Cfrac%7B%5Cbar%20x-%5Cmu%7D%7B%5Csigma%2F%5Csqrt%7Bn%7D%7D)
![=\frac{84-80}{3/\sqrt{3}}\\\\=2.31](https://tex.z-dn.net/?f=%3D%5Cfrac%7B84-80%7D%7B3%2F%5Csqrt%7B3%7D%7D%5C%5C%5C%5C%3D2.31)
The <em>z</em>-score for the mean blood pressure measurement of 3 patients is more than 2.
Thus, it would be unusual.