Human Memory and Sentence Recall: A Z-Test for Proportions

a. Hypothesis: People will perform significantly better than chance on the test sentence, indicating that they have retained the meaning of the text.

Null hypothesis: The proportion of people who correctly identify the test sentence is equal to chance (50%).

Alternative hypothesis: The proportion of people who correctly identify the test sentence is greater than chance (50%).

Test statistic: One-sample z-test for proportions.

Parameter: p, the proportion of people who correctly identify the test sentence in the population.

b. Sample proportion, p̂ = 27/45 = 0.6

Under the null hypothesis, the expected proportion is 0.5.

Standard error:

SE = sqrt(p̂(1-p̂)/n) = sqrt(0.5(1-0.5)/45) = 0.102

Z-score:

z = (p̂ - 0.5) / SE = (0.6 - 0.5) / 0.102 = 0.98

Using a standard normal table or calculator, the p-value for a one-sided test is 0.1635.

At the 10% level, we fail to reject the null hypothesis because the p-value is greater than 0.10.

At the 5% level, we fail to reject the null hypothesis because the p-value is greater than 0.05.

At the 1% level, we reject the null hypothesis because the p-value is less than 0.01.

Therefore, we can conclude that there is some evidence to suggest that people perform better than chance on the test sentence, but the evidence is not strong enough to reject the null hypothesis at the 10% or 5% level of significance.