Fifty two of a test panel of 175 consumers say that they would buy cereal A if it is presented on the market and 35 of another test panel of 150 consumers say that they would buy cereal B. Test the null hypothesis that the cereals on the market would be equally popular. Use a 5% significance level. Calculate the test statistic

Answer 1

To test the null hypothesis that the cereals on the market would be equally popular, we can use the chi-squared test.

First, we need to calculate the expected values for each category under the assumption of equal popularity.

The expected value for cereal A can be calculated as (total number of consumers)*(proportion of consumers who would buy cereal A).

Expected value for cereal A = 175 * (52/175) = 52

Similarly, the expected value for cereal B can be calculated as (total number of consumers)*(proportion of consumers who would buy cereal B).

Expected value for cereal B = 150 * (35/150) = 35

Next, we calculate the chi-squared test statistic using the formula:

chi-squared = Σ((observed value - expected value)^2 / expected value)

In this case, we only have two categories (cereal A and cereal B), so the formula simplifies to:

chi-squared = ((observed value for cereal A - expected value for cereal A)^2 / expected value for cereal A) + ((observed value for cereal B - expected value for cereal B)^2 / expected value for cereal B)

Let's plug in the values:

chi-squared = ((52 - 52)^2 / 52) + ((35 - 35)^2 / 35)
= 0 + 0
= 0

The test statistic is 0.

Now we need to determine the critical value for the chi-squared test at a 5% significance level, with a degree of freedom of 1 (number of categories - 1).

Using a chi-squared table or calculator, we find that the critical value at a 5% significance level with 1 degree of freedom is approximately 3.8415.

Since the test statistic (0) is less than the critical value (3.8415), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the cereals on the market are not equally popular.