A random sample of 200 people in southwestern Pennsylvania indicated that 88 were baseball fans. At is the proportion greater than 36%?

Question

1.)state the hypotheses and identify the claim

2.)compute the test value (test statistic).

3.)find the P-value

4.)make the decision

5.)summarize the results

Answer 1

First thing to do is write down your knowns:

a) It's a simple random sample (SRS)

i) n=200

ii) p_hat = 88/200 = 0.44

iii) check the conditions: (1) n*p_hat ≥ 30 is 200*0.44 = 88 which is greater than 30; and

(2) n*(1- p_hat) is 200*0.56 = 112 which is greater than 30. Both conditions and the SRS are met.

b) The claim is that the population proportion of fans is 36% (p=0.36).

c) Using a one-tailed test your null hypothesis is that the population proportion (p) is equal to 36%: The alternative hypothesis is that the true proportion is greater than 0.36. Thus:

H0: p = 0.36 (that the proportion is equal to 0.36)

Ha: p > 0.36 (that the proportion is greater than 0.36)

2) Now we compute the test statistic. Since the conditions are met and this is a proportion we standardize it to a z-score:

3) This means that the sample proportion of 44% is 2.38 standard deviations above the mean of all sample proportions (if the true mean were 36%). Now to find the p-value we use the z-table. I’m using this one here: http://www.z-table.com/

We see that at 2.83 approximately 0.9977 (99.77%) of sample proportions are below it. Thus, the probability of getting a sample with a p_hat >= 0.44 is (1-0.9977=0.0023). Therefore there is only a 0.23% chance of getting a sample with the proportion of 0.44 or higher if the true population proportion were really 0.36.

4) Since the p-value < 0.05 (0.0023 < 0.05), we have sufficient evidence to reject the null hypothesis (reject H0).

5) The likelihood of getting a random sample with a proportion of 0.44 is less than 5%. With this evidence we reject the null hypothesis that the true population proportion is 36%.

Let me know if you have any questions!!