Question

You flip a fair coin N=100 times. Approximate the probability that the proportion of heads among 100 coin tosses is at least 45%.

Question 4. You conduct a two-sided hypothesis test (α=0.05): H0: µ=25. You collect data from a population of size N=100 and compute a test statistic z = - 1.5. The null hypothesis is actually false and µ=22. Determine which of the following statements are true.

I) The two-sided p-value is 0.1336.

II) You reject the null hypothesis H0.

III) You fail to reject H0 and make a Type II error.

The answer for the first is 84% and the second is II and III only. I just want to know why its those. Thank you

Answer 1

N = 100

P[ head ] = 50% = 0.5 ( theoritical probability )

variance of p = p*(1-p)/n = 0.5*0.5/100 = 0.0025

standard deviation, SD = sqrt(variance ) = 0.05

we need to find probability of p being greater than 45%, where p follows N(0.5,0.0025)

P[ p > 0.45 ] = P[ ( p - 0.5 )/0.05 > ( 0.45 - 0.5 )/0.05 ] = P[ Z > -1 ] = 1 - P[ Z < -1 ] = 1 - 0.16 = 0.84

Probability of getting at least 45% heads in 100 trials is 45%

4. H0 : mu = 25

H1 : mu = 22 ( actually false )

N = 100

alpha = 0.05

z = -1.5

p value for two sided = 0.8664 ( from normal table )

reject H0 if p < alpha , here p ( 0.8664 ) > alpha ( 0.05 ), we failed to reject H0 and accept H1

and since, it is given that h1 is false, we accepted false H1 hence, made type II error.

The only correct option is III