Question

Someone esimates that the ratio of fans who are over 50 years old to fans who are under 50 years old is 3 to 1 in a basketball game. You think he’s exaggerating and that the ratio is actually lower. The next time you attend the arena game, you plan to watch 100 people enter the arena and count the number who are over 50 out of that 100. How many (or, perhaps, how few) over 50 fans would you have to count to reject your friend’s hypothesis of a 3 to 1 ratio (at α = .05)and conclude in favour of your idea that there are fewer over 50 fans than your friend thinks?

Answer 1

Let o50 represent over 50 and y50 represent younger than 50

As per your friend, o50/y50 = 3/1, that is out of 100 fans, 75 will be o50 and 25 will be y50

As per you, o50/y50 < 3/1, that is out of 100 fans, less than 75 will be o50 and more than 25 will be y50

Thus, Ho: p = 0.75, while according to you Ha: p < 0.75

α = 0.05

Critical z- score for a left-tailed test = -1.645

We want to find p’ such that you are right, that is, we want to find p’ such that we can reject Ho

Standard error SE = √{p(1 – p)/n} = √{0.75 * 0.25/100} = 0.0433

Test statistic z = (p’ – p)/SE = (p’ – 0.75)/0.0433

(p’ – 0.75)/0.0433 < -1.645

Upon solving, we get p’ < 0.6788

This means you will have to count fewer than 0.6788 * 100, that is fewer than 68 fans who are over 50 years to prove your friend wrong.

[Please give me a Thumbs Up if you are satisfied with my answer. If you are not, please comment on it, so I can edit the answer. Thanks.]