means to say that for any given , we can find
such that anytime
(i.e. the whenever
is "close enough" to 5), we can guarantee that
(i.e. the value of
is "close enough" to the limit value).
What we want to end up with is
Dividing both sides by 3 gives
which suggests is a sufficient threshold.
The proof itself is essentially the reverse of this analysis: Let be given. Then if
and so the limit is 7. QED
"β", the probability of a type II error here in this question is, the electrician cannot conclude that more than 90% of homes in the city are up to the current electric codes when, in fact, more than 90% of homes are up to the current electric codes.
As per the question statement, the null hypothesis, , is that, no more than 90% of homes in the city are up to the current electric codes, while an alternative hypothesis,
, is by an electrician claiming more than 90% of homes in the city are up to the current electric codes.
We are required to find out "β" which is the probability of a type II error.
β = Probability of accepting when it is incorrect.
The probability of a Type II error occurs in the situation where, the null hypothesis of "more than 90% of homes in the city are up to the current electric codes" is accepted even though the alternative hypothesis that the electrician claims about "more than 90% of homes in the city being up to the current electric codes" is true. More clearly, the alternative hypothesis that the electrician claims about more than 90% of homes in the city being up to the current electric codes is in fact, rejected when it is correct.
Hence the correct answer :
The probability of Type II error is (β) is that, "The electrician cannot conclude that more than 90% of homes in the city are up to the current electric codes when, in fact, more than 90% of homes are up to the current electric codes".
To learn more about Null & Alternative Hypothesis, click on the link below.
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