Chi Square - Goodness of Fit

Question

A psychic claims to be able to predict the future. You test the psychic's claim by flipping a quarter 20 times - the psychic guesses whether it will land heads or tails. You'd expect the psychic to be right 50% of the time just by chance (10 times right, 10 times wrong). The psychic predicts the coin correctly 14 times and is wrong 6 times. What do we conclude about the psychic's ability - can the psychic predict the future or is the psychic's performance consistent with chance guessing?

Null hypothesis:

Research hypothesis:

Decide whether to reject the null hypothesis. What do the results mean?

Answer 1

H₀: The sample data follow the hypothesized distribution (the psychic will guess 10 correct, 10 incorrect).

H₁: The sample data do not follow the hypothesized distribution (the psychic will guess more or fewer than 10).

Calculate the Chi-square value. Chi-square = (Observed - Expected)²/Expected for all data summed. (14-10)²/10 + (6-10)²/10 = 16/10 + 16/10 = 3.2.

Degrees of freedom = Number categories (heads, tails) - 1 = 1.

Use a table of critical values at 1 df to see that 3.84 is the chi-square value with a 5% chance of obtaining. Because our value is lower than this, we fail to reject the null. This means that the psychic in all likelihood was just guessing luckily.