Section 9.2 Hypothesis Tests for a Population Mean, Standard Deviation Known 413 The method we have described requires certain assumptions, which we now state. Assumptions for Performing a Hypothesis Test About �� When �� Is Known 1. We have a simple random sample. 2. The sample size is large (n > 30), or the population is approximately normal. When these assumptions are met, a hypothesis test can be performed using the following steps. Performing a Hypothesis Test for a Population Mean with �� Known Using the Critical Value Method Check to be sure the assumptions are satisfied. If they are, then proceed with the following steps. Step 1: State the null and alternate hypotheses. The null hypothesis specifies a value for the population mean ��. We will call this value ��0. So the null hypothesis is of the form H0 : �� = ��0. The alternate hypothesis can be stated in one of three ways: Left-tailed: H1: �� < ��0 Right-tailed: H1: �� > ��0 Two-tailed: H1: �� ≠ ��0 Step 2: Choose a significance level �� and find the critical value or values. Step 3: Compute the test statistic z = ̄x − ��0 ��∕ √ n . Step 4: Determine whether to reject H0, as follows: Left-tailed: H1: �� < ��0 Reject if z ≤ −z��. Right-tailed: H1: �� > ��0 Reject if z ≥ z��. Two-tailed: H1: �� ≠ ��0 Reject if z ≥ z��∕2 or z ≤ −z��∕2. Step 5: State a conclusion. EXAMPLE 9.8 Performing a hypothesis test with the critical value method The American Automobile Association reported that the mean price of a gallon of regular grade gasoline in the city of Los Angeles in July 2013 was $4.04. A recently taken simple random sample of 50 gas stations in Los Angeles had an average price of $3.99 for a gallon of regular grade gasoline. Assume that the standard deviation of prices is $0.15. An economist is interested in determining whether the mean price is less than $3.99. Use the critical value method to perform a hypothesis test at the �� = 0.05 level of significance. Solution We first check the assumptions. We have a simple random sample, the sample size is large (n > 30), and the population standard deviation �� is known. The assumptions are satisfied. Step 1: State H0 and H1. The null hypothesis says that the mean price is $4.04. Therefore, we have H0 : �� = 4.04 We are interested in knowing whether the mean price is less than $4.04. Therefore, the alternate hypothesis is H1: �� < 4.04 At this point, we assume H0 to be true.
navidi_monk_elementary_statistics_2e_ch7-9
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