Responses to a survey question about color preference for a candy are broken down according to gender in the table given below. At the 0.05 significance level, test the claim that candy color preference and gender are independent. $$\begin{array} { l | l l l }
& \text { Red } & \text { Blue } & \text { Yellow } \
\hline \text { Male } & 25 & 50 & 15 \
\text { Female } & 20 & 30 & 10
\end{array}$$ What is your conclusion about the null hypothesis and about the claim?
A) Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
B) Fail to reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
C) Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
D) Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.

Question

Responses to a survey question about color preference for a candy are broken down according to gender in the table given below. At the 0.05 significance level, test the claim that candy color preference and gender are independent. $$\begin{array} { l | l l l } 
& \text { Red } & \text { Blue } & \text { Yellow } \
\hline \text { Male } & 25 & 50 & 15 \
\text { Female } & 20 & 30 & 10
\end{array}$$ What is your conclusion about the null hypothesis and about the claim?
A) Reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
B) Fail to reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
C) Fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.
D) Reject the null hypothesis. There is sufficient evidence to warrant rejection of the claim that candy color preference and gender are independent.

Quizplus · Accepted Answer

We will use a chi-square test of independence. The expected counts for each cell if candy color preference and gender are independent are: $$\begin{array} { l | l l l | l } 
& \text { Red } & \text { Blue } & \text { Yellow } & \text{Total}\
\hline \text { Male } & 22 & 44 & 14 & 80\
\text { Female } & 23 & 36 & 11 & 70\
\hline \text{Total} & 45 & 80 & 25
\end{array}$$ The formula for the test statistic is: $$\chi^2 = \sum \frac{\left( O - E \right)^2}{E}$$ where O represents the observed frequency and E represents the expected frequency. Calculating the test statistic: $$\begin{array}{crrrrrr} \chi^2 &=& \frac{\left( 25 - 22 \right)^2}{22} &+& \frac{\left( 50 - 44 \right)^2}{44} &+& \frac{\left( 15 - 14 \right)^2}{14} \ &&+& \frac{\left( 20 - 23 \right)^2}{23} &+& \frac{\left( 30 - 36 \right)^2}{36} &+& \frac{\left( 10 - 11 \right)^2}{11} \ &=& 0.455 &+& 1.0227 &+& 0.0714 &+& 0.4565 &+& 1.5 &+& 0.0909 \ &=& 3.5965 \end{array}$$ Using software or a chi-square distribution table with 2 degrees of freedom, the critical value at the 0.05 level of significance is 5.99. Since the calculated test statistic is less than the critical value, we fail to reject the null hypothesis. We do not have sufficient evidence to reject the claim that candy color preference and gender are independent. Therefore, the best choice is C.

Responses to a Survey Question About Color Preference for a Candy