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In statistical testing, the null hypothesis is a statement we make about a population parameter that we will test using sample data. It is often denoted as \( H_0 \). For the chi-square test, the null hypothesis typically asserts that there is no significant difference between observed and expected frequencies.
In the context of our exercise, the null hypothesis states that the number of boxes sold for each packaging color is the same. This means that any differences we see in the data are due to random chance rather than the color of the box having a real effect.
We approach this by comparing the data against what we would expect if the null hypothesis were true. By doing this, we can see if the observed differences are larger than we would expect by chance alone. The chi-square test helps us quantify this comparison.

The significance level, denoted as \( \, \alpha \, \), is the probability threshold we set to decide whether to reject the null hypothesis. It represents how much risk we are willing to take of making a Type I error, which is rejecting a true null hypothesis.
In our exercise, the significance level is set at 1% or 0.01. This is quite a stringent level, meaning we require strong evidence to reject the null hypothesis.

• If the p-value we calculate is less than or equal to the significance level, we reject the null hypothesis.
• If the p-value is higher than the significance level, we do not reject the null hypothesis.

By setting this threshold, we are defining how certain we must be before concluding that the color affects sales. A lower \( \, \alpha \, \) reduces the chance of a Type I error but increases the chance of a Type II error, which is failing to reject a false null hypothesis.

Observed frequencies are the actual counts we obtain from our sample data. These are the numbers recorded in each category – for instance, the number of boxes sold for each color in our packaging exercise.
Expected frequencies, on the other hand, are the counts we would anticipate if the null hypothesis were true. These are calculated by evenly distributing the total number of observations across all categories. In our example, the expected frequency was 278.8 boxes for each color (total sales of 1394 divided by 5 colors).
The chi-square test statistic is calculated by comparing these two sets of frequencies. The calculation involves finding the sum of the squared differences between observed and expected frequencies, divided by the expected frequency for each group. This helps us understand whether the discrepancies we see are likely due to random variation or not.

Degrees of freedom (df) in a chi-square test refers to the number of values that are free to vary when computing a statistic. It is an important concept as it influences the distribution used for determining the p-value.
In the context of our chi-square test for independence, degrees of freedom are calculated by subtracting one from the number of categories in the data. For our exercise, where we have five different box colors, the degrees of freedom are calculated as:

• \( \text{df} = 5 - 1 = 4 \)

The degrees of freedom help determine the chi-square distribution that our test statistic is compared against. A higher number of categories generally increases the degrees of freedom, spreading the distribution. This aspect ensures the test adapts to the structure and size of the problem being analyzed.