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
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.
To test the null hypothesis that the cereals on the market would be equally popular, we can use the chi-squared test of independence. Let's calculate the test statistic step-by-step:
Step 1: Set up the hypotheses:
Null Hypothesis (H0): The cereals on the market are equally popular.
Alternative Hypothesis (HA): The cereals on the market are not equally popular.
Step 2: Determine the significance level:
The significance level is given as 5%, which corresponds to a significance level of 0.05.
Step 3: Calculate the expected values:
To test for independence, we need to determine the expected values for each category if the null hypothesis is true.
Expected count for cereal A = (proportion of cereal A buyers) * Total number of participants
Expected count for cereal A = (52/175) * 325 = 97.828
Expected count for cereal B = (proportion of cereal B buyers) * Total number of participants
Expected count for cereal B = (35/150) * 325 = 75.833
Step 4: Calculate the test statistic:
The chi-squared test statistic formula for independence is:
χ² = Σ((Observed count - Expected count)² / Expected count)
Using the observed and expected counts for both cereals from the given data, we can calculate the test statistic as follows:
χ² = ((52-97.828)² / 97.828) + ((35-75.833)² / 75.833)
Step 5: Calculate the degrees of freedom:
The degrees of freedom for a chi-squared test in this case is given by (number of categories - 1):
Degrees of freedom = (2 - 1) = 1
Step 6: Look up the critical value:
At a significance level of 0.05 and with 1 degree of freedom, the critical value of the chi-squared distribution is approximately 3.841.
Step 7: Compare the test statistic to the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
If the calculated test statistic is greater than the critical value of 3.841, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
I will calculate the test statistic for you.