Example

Distribution of Color in Plain M&M Candies

According to the manufacturer of M&M candy, the color distribution for plain chocolate M&Ms is 13% brown, 13% red, 14% yellow, 24% blue, 20% orange, 16% green. This statement about the distribution of color in plain M&Ms is the null hypothesis. The alternative hypothesis says that this is not the distribution.

• H₀: The color distribution for plain M&Ms is 13% brown, 13% red, 14% yellow, 24% blue, 20% orange, 16% green.
• H_a: The color distribution for plain M&Ms is different from the distribution stated in the null hypothesis.

We select a random sample of 300 plain M&M candies to test these hypotheses. If the sample has the distribution of color stated in the null hypothesis, then we expect 13% of the 300 to be brown, 13% of 300 to be red, 14% of 300 to be yellow, 24% of 300 to be blue, and so on. Here are the expected counts of each color for a sample of 300 candies:

Of course, the distribution of color will vary in different samples, so we need to develop a way to measure how far a sample distribution is from the null distribution, something analogous to a z-score or T-score. Before we discuss this new measure, let’s look at two random samples selected from the null distribution to practice recognizing different amounts of variability. We can compare the distributions visually using ribbon charts.

Which random sample deviates the most from the null distribution? We address this question in the next activity.

Example

Calculating χ2

For Sample 1, the chi-square test statistic is approximately 12.94. For Sample 2, the chi-square test statistic is approximately 1.53. Usually, we use technology to calculate χ², but here we show two calculations in detail to illustrate how the formula works. Notice that we are adding six terms. Each term represents the deviation for one color category.

Random Sample #1 (n=300):

[latex]\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{brown}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{red}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{yellow}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{blue}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{orange}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{green}\\ {\mathrm{χ}}^{2}=\frac{{(56-39)}^{2}}{39}+\frac{{(44-39)}^{2}}{39}+\frac{{(47-42)}^{2}}{42}+\frac{{(62-72)}^{2}}{72}+\frac{{(53-60)}^{2}}{60}+\frac{{(38-48)}^{2}}{48}\\ \text{}\text{}\text{}\text{}=\text{}\text{}\text{}\text{}\text{}\frac{289}{39}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\frac{25}{39}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\text{}\frac{25}{42}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\frac{100}{72}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\frac{49}{60}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\frac{100}{48}=\\ \text{}\text{}\text{}\text{}\text{}\text{}\approx \text{}\text{}\text{}\text{}7.41\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}0.64\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}0.60\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}1.39\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}0.82\text{}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\text{}2.08=12.94\text{}\end{array}[/latex]

Comment: In Sample 1, notice that both blue and green observed counts deviate from the expected counts by 10 candies. But green contributes more to the chi-square test statistic. This makes sense because the chi-square test statistic measures relative difference. Relative to the expected count of 48 green candies, an absolute error of 10 is large. It is almost 20% of the expected count. (10/48 is about 0.20). The squared difference relative to the expected count is 100/48, about 2.08. Relative to the expected count of 72 blue candies, an error of 10 candies is smaller. It is only about 14% of the expected count (10/72 is about 0.14). The squared difference relative to the expected count is 100/72, about 1.39.

Here is the chi-square calculation for Sample 2.

Random Sample #2 (n=300):

[latex]\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{brown}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{red}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{yellow}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{blue}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{orange}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\mathrm{green}\\ {\mathrm{χ}}^{2}=\frac{{(39-39)}^{2}}{39}+\frac{{(34-39)}^{2}}{39}+\frac{{(38-42)}^{2}}{42}+\frac{{(76-72)}^{2}}{72}+\frac{{(64-60)}^{2}}{60}+\frac{{(49-48)}^{2}}{48}\\ \text{}\text{ }=\text{}\text{}\text{}\text{}\text{}\frac{0}{39}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\frac{25}{39}\text{}\text{}\text{}\text{}\text{}\text{ }+\text{}\text{}\text{}\text{}\text{}\text{}\text{}\frac{16}{42}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\frac{16}{72}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\frac{16}{60}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\frac{1}{48}=\\ \text{}\text{}\text{}\text{}\text{}\text{}\text{}\approx \text{}\text{}\text{}\text{}0\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}0.64\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}0.38\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}0.22\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}0.27\text{}\text{}\text{}\text{}\text{}\text{}\text{}+\text{}\text{}\text{}\text{}\text{}\text{}\text{}0.02=1.53\text{}\end{array}[/latex]