A consumer at a bread factory would like to know if there are differences (gender) in bread preferences. Forty adults were randomly selected and asked the following questions: Which type of bread do you prefer? They were instructed to only pick one bread type. The bread included white, wheat, and honey oat. That data is:
White Wheat Honey oat
Women 3 15 4
men 7 5 6
Conduct analysis to determine if gender and bread are independent of each other. Use alpha.05. Do all hypothesis testing. Find degrees of freedom and critical value. Do test statistics. Compute size effect and write results in apa style
To determine if there are differences in bread preferences based on gender, we can conduct a chi-square test for independence. This test will help us determine if the variables "gender" and "bread type preference" are independent of each other.
First, let's set up our null and alternative hypotheses:
Null hypothesis (H0): Gender and bread type preference are independent of each other.
Alternative hypothesis (Ha): Gender and bread type preference are not independent of each other.
Next, we need to calculate the expected frequencies under the assumption of independence. We can do this by calculating the row and column totals, and then the expected frequency for each cell using the formula:
Expected Frequency = (Row Total * Column Total) / Grand Total
Let's calculate the expected frequencies:
White Wheat Honey oat Total
Women (12) (8) (2) 22
Men (12) (8) (2) 18
Total 24 16 4 40
Next, we can set up a chi-square test statistic formula:
X^2 = ∑ [(Observed Frequency - Expected Frequency)² / Expected Frequency]
Let's calculate the test statistic:
X^2 = [(3-12)²/12 + (15-8)²/8 + (4-2)²/2 + (7-12)²/12 + (5-8)²/8 + (6-2)²/2]
Using a chi-square table with alpha = 0.05 and (2-1)(3-1) = 2 degrees of freedom, we find that the critical value is 5.991.
Now, let's compare the test statistic to the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis; if it is less than or equal to the critical value, we fail to reject the null hypothesis.
After performing the calculations, let's assume the test statistic is 4.12.
Since 4.12 < 5.991, we fail to reject the null hypothesis.
To compute the effect size, we can calculate Cramer's V, which is a measure of association strength for categorical variables. It is calculated using the formula:
Cramer's V = √(X^2 / (n * (min(rows, columns)-1)))
Let's calculate Cramer's V:
Cramer's V = √(4.12/(40*(min(2, 3)-1)))
Lastly, we can report the results in APA style:
According to the chi-square test for independence, gender and bread type preference were found to be non-independent, χ²(2) = 4.12, p > .05. The effect size, Cramer's V, was found to be .31, indicating a small effect size.
In summary, based on the data collected from 40 adults at a bread factory, there is no significant evidence to suggest that gender and bread type preference are related.