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 need to conduct a chi-square independence test. This test will help us determine if the observed differences in bread preferences are statistically significant or if they could have occurred by chance.

First, let's set up our hypotheses:

Null hypothesis (H0): Gender and bread preference are independent of each other.
Alternative hypothesis (H1): Gender and bread preference are dependent.

To conduct the chi-square test, we need to create an observed frequency table based on the given data:

White Wheat Honey oat
Women 3 15 4
Men 7 5 6

Now, let's calculate the expected frequencies if there was no association between gender and bread preference. We can use the formula:

Expected frequency = (row total * column total) / grand total

Grand Total = 40 (sum of all observations)

Expected table:

White Wheat Honey oat
Women 6 9.5 4.5
Men 4 6.5 4.5

Next, we use the formula for the chi-square test statistic:

χ² = ∑ [(Oij - Eij)² / Eij]

Where Oij is the observed frequency and Eij is the expected frequency.

Calculating χ²:

χ² = [(3 - 6)² / 6] + [(15 - 9.5)² / 9.5] + ... + [(6 - 4.5)² / 4.5]

After calculating χ², we need to find the degrees of freedom (df) for the chi-square test. The degrees of freedom can be calculated using the formula:

df = (rows - 1) * (columns - 1)

In this case, df = (2 - 1) * (3 - 1) = 2.

Next, we need to find the critical value for the chi-square test at a significance level of 0.05. We can use a chi-square distribution table or a statistical software to find this value. With df = 2 and α = 0.05, the critical value is approximately 5.991.

Now, we compare the calculated chi-square value to the critical value. If the calculated chi-square value 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.

Finally, to compute the effect size, we can use Cramer's V coefficient, given by the formula:

Cramer's V = sqrt(χ² / (n * (min(rows - 1, columns - 1))))

Where n is the total sample size.
Substituting the values into the formula, we can calculate Cramer's V.

Once we have the results, we can report them in APA style.