Determine the P-value of the hypothesis test performed in Problem 9.

Data from Problem 9

The median salary of a teacher in Illinois is $57,283 according to TeacherSalaryInfo.com. The state of Illinois is experiencing serious budget issues in its educational institutions. A politician obtains the following data based on a random sample of 12 teachers at a district in Illinois currently experiencing financial difficulty. Do the data support the belief that the teachers in the district are paid less than the average teacher in the state? Use the a = 0.05 level of significance. Note: The data are skewed right with an outlier.

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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53,620 40,836 48,152 92,006 49,893 48,876 45,470 73,316 56,838 69,899 47,508 113,924