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Testing Claims about a Population Mean Page 1 Testing a Claim Regarding a Population Mean Step 0: Verify Assumptions The hypothesis test of a population mean has two assumptions. 1. The sample is obtained using simple random sampling 2. The sample has no outliers and the population from which the sample is drawn is normally distributed or the sample size, n, is large (n ≥ 30). Step 1: State the Hypothesis A claim is made regarding the population mean. This claim is used to determine the null and alternative hypotheses. The hypotheses can be structured in one of the following ways: Two-Tailed H0: µ = µ0 H1: µ ≠ µ0 Left-Tailed H0: µ ≥ µ0 H1: µ < µ0 Right-Tailed H0: µ ≤ µ0 H1: µ > µ0 Step 2: Select a Level of Significance The selection of the level of significance α is done based on the seriousness of making a Type I error. (The typical value of α is 0.05.) Step 3: Calculate the Test Statistic The test statistic represents the number of standard deviations the sample mean is from the claimed population mean, µ0. When σ is known, then a z-value may be found. When σ is unknown, then a t-value must be found z-Test x − µ0 z0 = t-Test x − µ0 t0 = s n σ n Step 4 (The Classical Approach): Find the Critical Value The level of significance is used to determine the critical value, represented by the t-values in the figures below. The critical region includes the values of the shaded region. The shaded region is α. Two-Tailed −t α / 2 Robert A. Powers tα / 2 Left-Tailed −tα Right-Tailed tα University of Northern Colorado Testing Claims about a Population Mean Page 2 Step 4 (The Modern Approach): Find the p-Value Based on the critical value t0, determine the probability that a sample mean is further from the mean than is hypothesized. This is represented by the shaded region in the figures below. Two-Tailed P(T < -|t0| or T > |t0|) − | t0 | | t0 | Left-Tailed P(T < -t0) − t0 Right-Tailed P(T > t0) t0 Step 5: Make a Decision Reject the null hypothesis if the test statistics lies in the critical region or the probability associated with the test statistic is less than the level of significance. Do not reject the null hypothesis if the test statistic does not lie in the critical region or the probability associated with the test statistic is greater than or equal to the level of significance. Step 6: State the Conclusion State the conclusion of the hypothesis test based on the decision made and with respect to the original claim. Reject H0 Do Not Reject H0 Original Claim is H0 There is sufficient evidence (at the α level) to reject the claim that … . There is not sufficient evidence (at the α level) to reject the claim that … . Robert A. Powers Original Claim is H1 There is sufficient evidence (at the α level) to support the claim that … . There is not sufficient evidence (at the α level) to support the claim that … . University of Northern Colorado