Question

1) Make the best matching (i.e. each option gets exactly one partner) between the following terms/phrases and the terms/phrases/expressions/scenarios below: Type I error, Type II error, null hypothesis, alternative hypothesis, practical significance, statistical significance, strong evidence in favor of H1, point estimate, test statistic

(a) H0 is not rejected when it is, in fact, false.

(b) the p-value is 0.005

(c) z = 1.21

(d) p = 0.15

(e) ˆp = 0.15

(f) p 6= 0.5

(g) H0 is rejected when it is, in fact, true.

(h) your results favor the alternative hypothesis

(i) what the confidence interval can tell you 10.

2) Suppose that you have two events, A and B, where P(A) = 0.5 and P(B) = 0.6. The probability of at least one of these events occuring is 0.8. Given this information,

(a) Find P(A ∪ B).

(b) Find P(A ∩ B).

(c) Find P(AC ).

(d) Find P(B|A).

(e) Are A and B mutually exclusive events?

(f) Are A and B necessarily independent events?

Answer 1

1(a) Type II error

( Accepting a false null hypothesis is type II error)

	H0 true	H0 false
Reject H0	Type I error	Correct decision
Do not reject H0	Correct decision	Type II error

(b) P value is 0.005 : Strong evidence in favor of H1

Note : H1 is alternative hypothesis , If P value is the probability of getting observed sample value or more extreme if null hypoithesis is true . Thus if P value is less , then it means H1 is true .

(c) z =1.21 : Test statistic

Note : For z test for mean or proportion we use the test statistic z

(d) p =0.15 : Null hypothesis

Note : Null hypothesis is the assumption about the population parameter. p is the population proportion

(e) =0.15 : Point estimate

Note : Point estimate of p is the sample proportion

(f) p 0.5 : Alternative hypothesis

Note : It is the opposite statement of null hypothesis. Alternative hypothesis takes these sign : or > or <

(g) Type I error

Note : Type I error is the error of rejecting a true null hypothesis

(h) Statistical significance

Note : When we get significant result , we say it is in favor of alternative hypothesis

i) practical significance

Note : Confidence interval gives the idea of how much different the sample statistic from the null distribution

2(a) P( A U B) = 0.8

Note : P( A U B) = P( A) +P(B) -P( A B ) , which means atleast one of the events A and B occur

(b) P( A B ) = P( A ) +P(B) -P( A U B) = 0.5 +0.6-0.8 = 0.3

(c) P( A c) = 1-P(A) = 1-0.5 =0.5

(d) P( B I A ) = P( A B ) / P( A) = 0.3/ 0.5 = 0.6

(e) A and B are mutually exclusive if P( A B ) =0

P( A B ) = 0.3   0

A and B are not mutually exclusive

(f)A and B are independent if

P( A B ) = P( A ) *P( B)

P( A B ) = 0.3  

P( A) *P( B) = 0.3

A and B are independent .