Question

An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for one year. The fire department is concerned that many houses remain without detectors. Let p = the true proportion of such houses having detectors and suppose that a random sample of 25 homes is inspected. If the sample strongly indicates that fewer than 80% of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless sample evidence strongly argues for their necessity. Let X denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that p > .8 if x < 15, where x is the observed value of X.
a) what is the probability that the claim is rejected when the actual value of p is .8?
b) what is the probability of not rejecting the claim when the actual value of p is .8?
c) how do the error probabilities of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14?

Answer 1

Answer:

a. P=0.01222

b. P=0.98778

c. The probability of rejecting the claim is now P=0.00298.

Step-by-step explanation:

In this case, we evaluate the sampling distribution for a population proportion π=0.8 with a sample size of 25.

We need to calculate the probability of getting a sample mean below 15, which means p=15/25=0.6.

The standard deviation of the sampling distribution is:

$\sigma_M=\sqrt{\frac{\pi(1-\pi)}{N}} =\sqrt{\frac{0.8\cdot0.2}{25}} =\sqrt{\frac{0.16}{25} } =0.08$

The z value por p=0.6 is

$z=\frac{p-\pi+0.5/N}{\sigma} =\frac{0.6-0.8+0.5/25}{0.08}= \frac{-0.18}{0.08} =-2.25$

The probability of having a sample mean less than 15 is

$P(\bar X$

The probaiblity of not rejecting the claim is 1-0.01222=0.98778

If the value 15 is replaced by 14, we have a new value of p=14/25=0.56.

There will be less chances of rejecting the hypothesis.

$z=\frac{p-\pi+0.5/N}{\sigma} =\frac{0.56-0.8+0.5/25}{0.08}= \frac{-0.22}{0.08} =-2.75$

$P(\bar X$