It is theorized that the systolic blood pressure of executives at a top hedge fu

Question

It is theorized that the systolic blood pressure of executives at a top hedge fund company have high blood pressures due to stress at work. A random sample of 72 executives had their blood pressures measured. The hypotheses are H0 : µ = 128 and Ha : µ > 128. Assume that the population standard deviation is = 15, and we want to conduct a hypothesis test at 0.02 significance level. If the mean blood pressure of all executives is 133, what is the power of this hypothesis test? As you work through this question, be sure to address the following:

(a) What is the critical value, z , of this hypothesis test?

(b) What are the values of the sample mean that will lead to rejecting this hypothesis test?

****MY STRUGGLE***** (c) How likely are we to obtain values of the sample mean from the previous part if the mean blood pressure is truly 133? (This gives the power of the test).

I need some step by step instruction on how to figure out how likely we are to obtain the values of the sample mean from the previous part if the mean blood pressure is truly 133. It is meant to be a sort-of work backwards problem, but I'm so confused as to what steps should be taken and why? I understand that when we find the critical value, we know what area will lead to rejecting our null, but I don't understand the logic of C. I don't really understand how to solve for power. Please do a step by step, kindof dumbed down version for me. Thank you!

Explanation / Answer

Given that,
population mean(u)=128
standard deviation, =15
sample mean, x =133
number (n)=72
null, Ho: =128
alternate, H1: >128
level of significance, = 0.02
from standard normal table,right tailed z /2 =2.054
since our test is right-tailed
reject Ho, if zo > 2.054
critical value
the value of |z | at los 2% is 2.054
ANSWERS
---------------
a.
critical value: right tailed z /2 =2.054

b.
Given that,
Standard deviation, =15
Null, H0: =128
Alternate, H1: >128
Level of significance, = 0.02
From Standard normal table, Z /2 =2.0537
Since our test is right-tailed
Reject Ho, if Zo < -2.0537 OR if Zo > 2.0537
Reject Ho if (x-128)/15/(n) < -2.0537 OR if (x-128)/15/(n) > 2.0537
Reject Ho if x < 128-30.8055/(n) OR if x > 128-30.8055/(n)
-----------------------------------------------------------------------------------------------------
Suppose the size of the sample is n = 72 then the critical region
becomes,
Reject Ho if x < 128-30.8055/(72) OR if x > 128+30.8055/(72)
Reject Ho if x < 124.36953701 OR if x > 131.63046299
Implies, don't reject Ho if 124.36953701 x 131.63046299
Suppose the true mean is 133
Probability of Type II error,
P(Type II error) = P(Don't Reject Ho | H1 is true )
= P(124.36953701 x 131.63046299 | 1 = 133)
= P(124.36953701-133/15/(72) x - / /n 131.63046299-133/15/(72)
= P(-4.88212712 Z -0.77472713 )
= P( Z -0.77472713) - P( Z -4.88212712)
= 0.2193 - 0 [ Using Z Table ]
= 0.2193
For n =72 the probability of Type II error is 0.2193

c.
power of test = 1 - Type II error = 1 - 0.2193 = 0.7807

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