• P-values measure evidence against the the null hypothesis

• In sports, namely the NFL for this example, they help us ask: Was the outcome due to chance, or realized skill?

• Example: A kicker makes 18 out of 20 field goals in a season. Is he better than the average?

• Suppose the league average field goal (FG) percentage is 80%.

• Null Hypothesis: \(H_0 : player's accuracy = 0.80\)

• Alternative Hypothesis: \(H_a : player's accuracy > 0.80\)

• Observed Hypothesis: 18/20 = 90%

• Test Statistic: \[ \hat{p} = \frac{18}{20} = 0.9 \]

• Standard Error: \[ SE = \sqrt{\frac{0.8(1-0.8)}{20}} \approx 0.089 \]

• Z-statistic: \[ Z = \frac{0.9-0.8}{0.089} \approx 1.12 \]

P-value: \[ p = 1 - \Phi(1.12) \approx 0.13 \]

• Interpretation: The probability of seeing 90% (or higher) FG success if the kicker’s accuracy is true at 80% is 13%.

• Since the P-value is > 0.05, the result is not statistically significant enough to reject the null hypothesis.

n <- 20
x <- 18
p0 <- 0.8

phat <- x/n
SE <- sqrt(p0*(1-p0)/n)
Z <- (phat - p0)/SE
p_value <- 1 - pnorm(Z)
sprintf("P-value equals %.2f", p_value)

## [1] "P-value equals 0.13"

• An NFL kicker’s FG rate of 90% may seem impressive, but:

• P-value = 0.13 could easily happen by chance.

• With only 20 attempts, the sample size is not large enough to imply that he is better than league average.

• Larger sample size would provide us a more reliable conclusion.