A social psychologist wants to determine if socioeconomic status is related to game-playing preferences. A total of 50 children in a random sample were identified from families of low, middle, and high socioeconomic status, and each child was asked to select one of the following 4 games: Monopoly, Battleship, Life, or Connect Four. The psychologist performed a hypothesis test to test this association which yielded a test statistic of 5.0. What can we say about the p-value,
𝐻
0
, and the conclusion at the
𝛼
=
0.01
level of significance?

Question

A social psychologist wants to determine if socioeconomic status is related to game-playing preferences. A total of 50 children in a random sample were identified from families of low, middle, and high socioeconomic status, and each child was asked to select one of the following 4 games: Monopoly, Battleship, Life, or Connect Four. The psychologist performed a hypothesis test to test this association which yielded a test statistic of 5.0. What can we say about the p-value, 
𝐻
0
, and the conclusion at the 
𝛼
=
0.01
 level of significance?

BlackTom AI · Accepted Answer

Here is the step-by-step analysis of each answer option, with careful attention to the given test statistic and significance level. Option 1: 0.2 < p-value < 0.3; fail to reject H0; a significant evidence of an association between socioeconomic status and game preference. - This statement asserts a p-value in the 0.2–0.3 range. Given a test statistic of 5.0 for an association test (likely a chi-square test with (r-1)(c-1) = (3-1)(4-1) = 6 degrees of freedom), the p-value is typically well above 0.3, not in the 0.2–0.3 interval. Moreover, the phrase 'a significant evidence' contradicts the condition of failing to reject H0; if p > 0.01, we would not deem the result statistically significant. Therefore this option is inconsistent both with the p-value range and with the conclusion about significance. Option 2: p-value > 0.3; fail to reject H0; no evidence of an association between socioeconomic status and game preference. - With a chi-square statistic of 5.0 and 6 degrees of freedom, the p-value is typically well above 0.3—indeed, it is commonly around 0.6 to 0.7 or higher depending on the exact df. A p-value greater than 0.3 easily exceeds the alpha = 0.01 threshold, so we fail to reject H0. The conclusion 'no evidence of an association' follows from a non-significant result, which aligns with the standard interpretation of such a test. Option 3: 0.2 < p-value < 0.3; fail to reject H0; no evidence of an association between socioeconomic status and game preference. - Similar to Option 1, this claims a p-value in the 0.2–0.3 range. As noted earlier, the test statistic of 5.0 with the given degrees of freedom would not typically yield a p-value in that narrow range. Additionally, the combination of 'p-value in 0.2–0.3' and 'fail to reject H0' can be internally consistent, but the specific p-value claim is unlikely for the reported statistic, making this option dubious. Option 4: p-value < 0.01; reject H0; a significant evidence of an association between socioeconomic status and game preference. - This option asserts a highly significant result (p < 0.01). For a chi-square statistic of 5.0 with 6 df, the p-value is not that small; it is much larger than 0.01. Therefore this option is inconsistent with the observed test statistic, and it would lead to an incorrect conclusion of a significant association. Overall assessment: - The only option that coherently matches a relatively large p-value given the test statistic and that leads to failing to reject H0 at alpha = 0.01 is Option 2. The other options either misstate the p-value range, assert an incorrect conclusion about significance, or mischaracterize the relationship between the variables. Note on interpretation: - In a chi-square test of independence, a test statistic of 5.0 with df = (r-1)(c-1) = 6 typically yields a p-value well above 0.01, supporting the conclusion of no association. The practical reading is that socioeconomic status and game-playing preferences do not show a statistically detectable relationship in this sample under the 1% significance level.

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