The hypotheses for a contingency table question are always the same format. The Null hypothesis reflects the idea that there is no difference between the groups with respect to preference (i.e., independence). The alternative reflects the idea that there is a difference between the groups with respect to preference (i.e., dependence). There are two common methods to state the hypotheses:

Method 1

Let us examine the percent of players from each level who do purchase the add-on and do the analysis on them:

• $P_B$ = percent of bronze level players who purchase the add-on
• $P_S$ = percent of silver level players who purchase the add-on
• $P_G$ = percent of gold or higher level players who purchase the add-on

We assume, if whether or not they purchase the add-on is is independent of their level:

$H_0:  P_B= P_S= P_G \leftarrow$ percent who purchase the add-on is the same amongst all levels of players

$H_A:$ At least one of $P_B, P_S, P_G$ is not equal $\leftarrow$ percent who purchase the add-on is not the same amongst all levels of players.

Method 2:

Alternatively you can formulate the hypotheses this way:

$H_0$: Likelihood to purchase add-on is independent of player level.

$H_A$: Likelihood to purchase add-on is dependent on player level.

We will use the following formula for our expected values:

$$Exp_i = \frac{\text{Row Total}\times \text{Column Total}}{\text{Sample Size}}$$

Let us calculate the expected values within the table:

We will use the following formula to calculate the χ² test statistic:

$$\chi^2_{test} = \sum \frac{(obs – exp)^2}{exp}$$

Let us calculate the differences between the observed (actual) and expected (exp) values within the table:

This gives $\chi^2_{test} = 0.4545 + 0.5556 + 0.2832 + 0.3452 + 0.7428 + 0.9079 = 1.6507$

To determine the p-value, we first need to calculate the degrees of freedom:

$$\text{Degrees of Freedom} = df = (\text{#} rows – 1)\times (\text{#} columns – 1) = (3-1) \times (2-1) = 2 \times 1 = 2$$

For the number of rows and columns, we only count the rows and columns that include values (not the totals or headers).

We can now calculate the p-value using Excel’s CHISQ.DIST.RT() function:

$$\text{p-value} = \text{CHISQ.DIST.RT}(1.6507, 2) = 0.438083$$

Because the p-value = 0.4381 is greater than (>) the level of significance (5%), we cannot reject H₀.

Because we cannot reject H₀, there is not sufficient evidence to conclude that whether or not a player purchases the add-on is dependent on the player’s level. What does this mean for the analysts? They should not use the player’s level when attempting trying to decide who to target when promoting the add-on pack.

Let us now perform all of the calculations using Excel:

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