Determine the P-value of the hypothesis test performed in Problem 7.

Data from Problem 7

In 2002, the median pH level of the rain in Glacier National Park, Montana, was 5.25. A biologist thinks that the acidity of rain has decreased since then. (This would mean that the pH level of the rain has increased.) She obtains a random sample of 15 rain dates in 2011 and obtains the following data:

We can use the P-value approach when determining whether or not to reject the null hypothesis regarding a median by using the sign test. Recall that the P-value is the probability of observing a test statistic as extreme or more extreme than what was actually observed, under the assumption that the null hypothesis is true.

In the sign test, we assume that the median is M₀, so 50% of the data should be less than M₀ and 50% of the data greater than M₀. So we expect half of the data to result in minus signs and half of the data to result in plus signs. We can think of the data as a bunch of plus and minus signs that follow a binomial probability distribution with p = ½ if the null hypothesis is true. So the P-value is computed from the binomial probability formula, with X = k and n equal to the number of plus and minus signs:

P-value = P(X ≤ k) = _nC_k0.5^k(1 - 0.5)^n-k + _nC_k-10.5^k-1(1 - 0.5)^n-(k-1) + ∙ ∙ ∙ + _nC₀(1 - 0.5)ⁿ

For Example 1 in this section, the P-value is

P-value = P(X ≤ 8) = ₂₀C⁸ ∙ 0.5⁸ ∙ (1 - 0.5)^20-8 + ₂₀C₇ ∙ 0.5⁷ ∙ (1 - 0.5)¹³ + ∙ ∙ ∙ + ₂₀C₀(1 - 0.5)²⁰ = 0.2517

Because the P-value is greater than the level of significance, a = 0.05, we do not reject the null hypothesis. These binomial probabilities are easiest to compute with statistical software or a graphing calculator with advanced statistical features.

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