Question

A noted psychic was tested for ESP. The psychic was presented with 400 cards face down and asked to determine if each of the cards was one of four symbols: a star, cross, circle, or square. The psychic was correct in 120 cases. Based on chance alone, the expected probability that a card will be chosen correctly is ¼. Fill out the table below of observed and expected counts. Psychic’s Outcome Observed Expected Correct 120 Incorrect Total 400 400 Calculate χ² components. Psychic’s Outcome χ² components Correct Incorrect Total Is she really psychic? Check for evidence that the proportion of cards correctly identified by the psychic is significantly different than expected based on chance alone. Complete the table below.

Test type:
Null hypothesis:
Alt hypothesis:
Test-statistic:
p-value:
Conclusion:

Psychic’s Outcome χ² components

Correct

Incorrect

Total

Answer 1

Test type: Chi-Square Test for Homogeneity

Null hypothesis H0: Proportion of cards correctly identified by the psychic  is same as expected based on chance alone (1/4)

Alternative hypothesis H1: Proportion of cards correctly identified by the psychic  is different than  expected based on chance alone (1/4)

Expected count for correct cases = 400 * 1/4 = 100

Expected count for incorrect cases = 400 * 3/4 = 300

Observed count for correct cases = 120

Observed count for incorrect cases = 400 - 120 = 280

Psychic’s Outcome χ² components

Correct (120 - 100)² / 100 = 4

Incorrect  (280 - 300)² / 300 = 1.33

Total 4 + 1.33 = 5.33

Test-statistic: 5.33

Degree of freedom = Number of cases - 1 = 2 - 1 = 1

p-value = P(χ² > 5.33, df = 1) = 0.021

Since p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that proportion of cards correctly identified by the psychic is significantly different than expected based on chance alone (1/4)