Question

A marketing survey involves product recognition in New York and California. Of 203 New Yorkers surveyed, 49 knew the product while 72 out of 299 Californians knew the product. At the 2.5% significance level, use the critical value method to test the claim that the recognition rates are different in the states.

Enter the smallest critical value. (Round your answer to nearest hundredth.)

Answer 1

The smallest critical value for this test is 1.96 (rounded to the nearest hundredth).

Here's how to find the smallest critical value for this hypothesis test:

1. Define the hypotheses:

Null hypothesis (H0): The recognition rates in New York and California are the same.

Alternative hypothesis (H1): The recognition rates in New York and California are different.

2. Choose the appropriate test statistic:

Since we are comparing two independent proportions, the appropriate test statistic is the z-score.

3. Calculate the pooled proportion:

p_pooled = (49 + 72) / (203 + 299) = 0.122

4. Calculate the standard error:

SE = sqrt( p_pooled*(1-p_pooled) * (1/203 + 1/299) ) = 0.019

5. Determine the critical value:

We need a two-tailed test at the 2.5% significance level. Using a z-table or calculator, the critical value is approximately 1.96.

Therefore, the smallest critical value for this test is 1.96 (rounded to the nearest hundredth).