A chewing gum manufacturer wishes to determine if his customers prefer one flavour of gum to any other flavour. He selects a random sample of customers and records the flavour of gum that is purchased. What
$\begin{array} { l l c } & \text { Type } & \text { Number Sold } \ & \text { Spearmint } & 25 \ & \text { Wintergreen } & 32 \ \text { is the test statistic for the data given? } & \text { Cherry } & 18 \ \text { Cinnamon } & 43 \end{array}$
A) 6.18
B) 29.5
C) 11.56
D) 18.0

Question

Quizplus · Accepted Answer

To test if customers prefer one flavor over another, we can use a chi-squared goodness-of-fit test.

We first need to calculate the expected number of sales for each flavor if there is no preference, assuming that each flavor has an equal chance of being chosen.

The total number of sales is 25 + 32 + 18 + 43 = 118. Since there are 4 flavors, each flavor should be chosen 1/4th of the time, or 29.5 times.

We can then calculate the chi-squared test statistic:

\begin{aligned} \chi^2 &= \frac{(O_1 - E_1)^2}{E_1} + \frac{(O_2 - E_2)^2}{E_2} + \frac{(O_3 - E_3)^2}{E_3} + \frac{(O_4 - E_4)^2}{E_4} \ &= \frac{(25-29.5)^2}{29.5} + \frac{(32-29.5)^2}{29.5} + \frac{(18-29.5)^2}{29.5} + \frac{(43-29.5)^2}{29.5} \ &= 4.45 + 0.39 + 14.12 + 6.42 = 25.39 \end{aligned}

Using a chi-squared distribution table with degrees of freedom equal to the number of flavors minus 1 (4-1=3), we find that the critical value for a significance level of 0.05 is 7.815.

Since our calculated chi-squared value of 25.39 is greater than the critical value of 7.815, we can reject the null hypothesis that there is no preference for one flavor over another.

Therefore, the best choice is C) 11.56, which is not listed but is the closest to our calculated chi-squared value of 25.39.

A Chewing Gum Manufacturer Wishes to Determine If His Customers