Question

a sample of chess players (x1) and checker players (x2) were given an iq test. the results of the iq tests were analyzed with a t - test. the critical value (tcv) was 2.05. if there is no difference in iq between chess players and checker players, there is a 5% chance that the standardized difference (t) in scores between sample means is equal to or greater than 2.05. if there is no difference in iq between chess players and checker players, there is a 5% chance that the standardized difference (t) in iq scores between sample means for chess players and checker players is equal to 2.05. if there is no difference in iq between chess players and checker players, there is a 5% chance that the standardized difference (t) in iq scores between sample means for chess players and checker players is equal to or greater than 2.05. regardless of difference in population means, there is a 5% chance that the standardized difference (t) between sample means is equal to or greater than 2.05. if there is a difference in iq between chess players and checker players, there is a 5% chance that the standardized difference (t) in scores between sample means is equal to 2.05.

Explanation:

In hypothesis testing with a t - test, the critical value is used to determine the rejection region. When the null hypothesis (no difference in IQ, in this case) is true, the critical value \(t_{cv}\) represents the value beyond which (in the tails of the t - distribution) we would reject the null hypothesis. A 5% significance level (common in many tests) means that if the null hypothesis is true, there is a 5% chance of getting a test statistic (the standardized difference \(t\) between sample means) as extreme or more extreme (equal to or greater than) the critical value.

The first option is incorrect because it says "equal to or greater than 2.05" but the critical value in a two - tailed test (implied here as we are comparing two groups without a prior directional hypothesis) for a 5% significance level actually accounts for both tails. However, if we assume a one - tailed test (not clearly stated, but the options are written in a way that we can analyze based on the concept), the second option is wrong because it says "equal to" (the critical value is a boundary, and the probability is for values equal to or beyond it). The fourth option is wrong because it ignores the null hypothesis (the critical value is calculated under the assumption of the null hypothesis, i.e., no difference in population means). The fifth option is wrong because it assumes there is a difference (critical value is calculated under the null hypothesis of no difference).

The third option is correct as it correctly states that under the null hypothesis (no difference in IQ between chess and checker players), there is a 5% chance that the standardized difference \(t\) (in IQ scores between sample means for chess and checker players) is equal to or greater than 2.05.

Answer:

The third option: If there is NO difference in IQ between chess players and checker players, there is a 5% chance that the standardized difference (t) in IQ scores between sample means for chess players and checker players is equal to or greater than 2.05.