Question

Suppose that in a random sample of 200 New York City residents 55% can name a player on the Knicks. In a random sample of 120 Toronto residents, 70% can name a player on the Raptors. At the 5% level of significance, determine if we can conclude that the proportion of all NYC residents that can name a player on the Knicks differs from the proportion of all Toronto residents who can name a player on the Raptors. C.

a. State the null and alternative hypotheses.
b. Are all criteria for the hypothesis test satisfied?

Answer 1

a) The null and alternative hypothesis are:

`H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\\eq 0`

b) Yes.

Explanation:

This is a hypothesis test for the difference between proportions.

The claim is that the proportion of all NYC residents that can name a player on the Knicks differs from the proportion of all Toronto residents who can name a player on the Raptors.

As the claim is that the proportions differs, the difference to reject the null hypothesis can be positive or negative. Then, this is a two-tailed test.

The null and alternative hypothesis are:

`H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\\eq 0`

The conditions to met for this test are:

- The sampling method for each population is simple random sampling.

- The samples are independent.

- Each sample includes at least 10 successes and 10 failures.

`\text{Failures (NY):} \, F=200*0.45=90\\\\\text{Failures (T):} \, F=120*0.3=36\\\\\text{Note: we use failures as they have the lower proportions.}`

- Each population is at least 20 times as big as its sample.

This conditions are met, so all criteria for the hypothesis test is satisfied.