Test Statistic & P-Value

Question

The proportion of people that own cats is 60%. A veterinarian believes that this proportion is smaller than 60% and surveys 800 people. Test the veterinarian's claim at the ∝ = 0.05 significance level.

Based on the sample of 800 people, 56% owned cats. Calculate the test statistic. Round to two decimal places.

Calculate the p-value. Round to four decimal places.

Answer 1

Hello Baile, To test the veterinarian's claim, you can perform a z-test for proportions. Here's how to calculate the test statistic and p-value:

1. Calculate the standard error of the sample proportion (p-hat):

SE = √[p-hat * (1 - p-hat) / n]

where p-hat is the sample proportion and n is the sample size.

SE = √[0.56 * (1 - 0.56) / 800]

SE ≈ 0.0329 (rounded to four decimal places)

2. Calculate the test statistic (z-score):

z = (p-hat - p) / SE

where p is the hypothesized population proportion (60% in this case).

z = (0.56 - 0.60) / 0.0329

z ≈ -1.2159 (rounded to four decimal places)

3. Calculate the p-value:

You'll need to find the z-table or use a calculator to find the cumulative probability associated with the z-score of -1.2159. The p-value corresponds to the probability that the sample proportion is as extreme as the one observed (56%) if the null hypothesis is true (population proportion is 60%).

4. The p-value for a z-score of -1.2159 can be found in a standard normal distribution table or calculated using a calculator or software. Once you find it, round it to four decimal places.

The final p-value will determine whether you can reject the null hypothesis at the 0.05 significance level. If the p-value is less than 0.05, you can reject the null hypothesis. If the p-value is greater than or equal to 0.05, you fail to reject the null hypothesis.