Question

Suppose we suspect a coin is not fair — we suspect that it has larger chance of getting tails than heads, so we want to conduct a hypothesis testing to investigate this question.

a: Let p be the chance of getting heads, write down the alternative hypothesis Ha and the null hypothesis H0 in terms of p.

b: In order to investigate this question, we flip the coin 100 times and record the observation. Suppose we use T = the number of heads as our test statistic, consider two potential rejection regions, 1. T ≥ 80 2. T ≤ 20 Which rejection region will help us reject H0 in favor of Ha?

c: Consider different cases where we observe T = 10, T = 90 and T = 50, what will be our corresponding conclusions given the choice of RR we made in part b?

Answer 1

The coin has larger chance of getting tails than heads, so we want to conduct a hypothesis testing to investigate.

a) Let p be the chance of getting heads, we are claiming that p < 0.5

Null hypothesis, Ho : chance of getting heads is exactly 0.5 equals to tails

or, Null hypothesis, Ho :  p = 0.5

Alternate hypothesis, Ha : chance of getting heads is less than 0.5

or, Alternate hypothesis, Ho : p < 0.5

b) Now, we flip the coin 100 times and record the observation.

we use T = the number of heads as our test statistic. Consider two potential rejection regions, 1. T ≥ 80 2. T ≤ 20 Which rejection region will help us reject H0 in favor of Ha?

H0 will be rejected if p < 0.5 or we get less number of heads than that of tails.

1. T ≥ 80 will help us reject H0 in favor of Ha since, in this case the chance of getting heads is less than 0.5 ( p < 0.5 )

c) Consider different cases where we observe T = 10, T = 90 and T = 50, what will be our corresponding conclusions given the choice of RR we made in part b?

T = 10 will made us reject null hypothesis that the probability of heads is 0.5 but it won't favor alternate hypothesis too because the probability of getting a tails is less than 0.5 ( 10 tails out of 100 )

T = 90 will made us reject null hypothesis that the probability of heads is 0.5 and it will favor alternate hypothesis too because the probability of getting a heads is less than 0.5 ( 10 heads out of 100 )

T = 50 will made us accept null hypothesis that the probability of heads is exactly 0.5 ( 50 tails out of 100 )