Answer:
a) Probability that the claim is rejected when the actual value of p is 0.8 = P(X ≤ 15) = 0.0173
b) Probability of not rejecting the claim when p = 0.7, P(X > 15) = 0.8106
when p = 0.6, P(X > 15) = 0.4246
c) Check Explanation
The error probabilities are evidently lower when 15 is replaced with 14 in the calculations.
Step-by-step explanation:
p is the true proportion of houses with smoke detectors and p = 0.80
The claim that 80% of houses have smoke detectors is rejected if in a sample of 25 houses, not more than 15 houses have smoke detectors.
If X is the number of homes with detectors among the 25 sampled
a) Probability that the claim is rejected when the actual value of p is 0.8 = P(X ≤ 15)
This is a binomial distribution problem
A binomial experiment is one in which the probability of success doesn't change with every run or number of trials (probability that each house has a detector is 0.80)
It usually consists of a number of runs/trials with only two possible outcomes, a success or a failure (we are sampling 25 houses with each of them either having or not having a detector)
The outcome of each trial/run of a binomial experiment is independent of one another.
Binomial distribution function is represented by
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
n = total number of sample spaces = 25 houses sampled
x = Number of successes required = less than or equal to 15
p = probability of success = probability that a house has smoke detectors = 0.80
q = probability of failure = probability that a house does NOT have smoke detectors = 1 - p = 1 - 0.80 = 0.20
P(X ≤ 15) = Sum of probabilities from P(X = 0) to P(X = 15) = 0.01733186954 = 0.01733
b) Probability of not rejecting the claim when p= 0.7 when p= 0.6
For us not to reject the claim, we need more than 15 houses with detectors, hence, th is probability = P(X > 15), but p = 0.7 and 0.6 respectively for this question.
n = total number of sample spaces = 25 houses sampled
x = Number of successes required = more than 15
p = probability that a house has smoke detectors = 0.70, then 0.60
q = probability of failure = probability that a house does NOT have smoke detectors = 1 - p = 1 - 0.70 = 0.30
And 1 - 0.60 = 0.40
P(X > 15) = sum of probabilities from P(X = 15) to P(X = 25)
When p = 0.70, P(X > 15) = 0.8105639765 = 0.8106
When p = 0.60, P(X > 15) = 0.42461701767 = 0.4246
c) How do the "error probabilities" of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14.
The error probabilities include the probability of the claim being false.
When X = 15
(Error probability when p = 0.80) = 0.0173
when p = 0.70, error probability = P(X ≤ 15) = 1 - P(X > 15) = 1 - 0.8106 = 0.1894
when p = 0.60, error probability = 1 - 0.4246 = 0.5754
When X = 14
(Error probability when p = 0.80) = P(X ≤ 14) = 0.00555
when p = 0.70, error probability = P(X ≤ 14) = 0.0978
when p = 0.60, error probability = P(X ≤ 14) = 0.4142
The error probabilities are evidently lower when 15 is replaced with 14 in the calculations.
Hope this Helps!!!
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(A)