Fifty two of a panal of 175 consumers say that they would buy cereal A if it is presented on the market and 35 of another test panel of 159 consumers say that they would buy cereal B.Test the bull hypothesis that the cereals on the market would be equally popular,use 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 for independence.
First, let's set up the hypothesis:
Null hypothesis (H0): The cereals on the market are equally popular.
Alternate hypothesis (Ha): The cereals on the market are not equally popular.
Next, we calculate the expected frequencies under the assumption that the cereals are equally popular. Since the null hypothesis assumes they are equally popular, we can assume an equal distribution of preference.
Expected frequency for Cereal A = (52/175) * 159 = 47.94
Expected frequency for Cereal B = (35/159) * 175 = 38.88
Now, we can calculate the test statistic:
χ^2 = Σ((Observed Frequency - Expected Frequency)^2 / Expected Frequency)
χ^2 = ((52 - 47.94)^2 / 47.94) + ((35 - 38.88)^2 / 38.88)
χ^2 ≈ 0.345
Finally, we can compare the test statistic to the critical value from the chi-squared distribution table at the 5% significance level and degrees of freedom equal to (number of rows - 1) * (number of columns - 1). Since we have a 2x2 table, the degrees of freedom is 1.
The critical value at the 5% significance level with 1 degree of freedom is approximately 3.841.
Since the test statistic (0.345) is less than the critical value (3.841), we fail to reject the null hypothesis. 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 a hypothesis testing approach. Let's assume that the proportion of consumers who would buy cereal A is denoted by p1, and the proportion who would buy cereal B is denoted by p2.
First, let's calculate the sample proportions:
p̂1 = 52/175 = 0.2971 (proportion of consumers who would buy cereal A)
p̂2 = 35/159 = 0.2208 (proportion of consumers who would buy cereal B)
Next, we can set up the null and alternative hypotheses:
Null Hypothesis (H0): p1 = p2
Alternative Hypothesis (H1): p1 ≠ p2
To test these hypotheses, we can use the z-test for comparing two proportions. The test statistic formula for this z-test is given by:
z = (p̂1 - p̂2) / √((p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2))
where n1 and n2 are the sample sizes of the two groups.
In this case, n1 = 175 and n2 = 159. Plugging in the values, we get:
z = (0.2971 - 0.2208) / √((0.2971 * (1 - 0.2971) / 175) + (0.2208 * (1 - 0.2208) / 159))
Calculating this expression, we obtain the test statistic z.
Note: The calculation of z requires manual computation, and the result may vary based on the level of precision used in the calculation.