448 Chapter 9 Hypothesis Testing NOTATION ∙ p is the population proportion of individuals who are in a specified category. ∙ p0 is the population proportion specified by H0. ∙ x is the number of individuals in the sample who are in the specified category. ∙ n is the sample size. ∙ ̂p is the sample proportion of individuals who are in the specified category. ̂p = x∕n. Explain It Again Reasons for the assumptions: The population must be much larger than the sample (at least 20 times as large), so that the sampled items are independent. The assumption that both np0 and n(1 − p0) are at least 10 ensures that the sampling distribution of p̂ is approximately normal when we assume that H0 is true. We can perform a test whenever the sample proportion ̂p is approximately normally distributed. This will occur when the following assumptions are met. Assumptions for Performing a Hypothesis Test for a Population Proportion 1. We have a simple random sample. 2. The population is at least 20 times as large as the sample. 3. The individuals in the population are divided into two categories. 4. The values np0 and n(1 − p0) are both at least 10. Either the critical value method or the P-value method may be used to perform a hypothesis test for a population proportion. We will present the steps for the P-value method first. Performing a Hypothesis Test for a Population Proportion Using the P-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 will have the form H0 : p = p0. The alternate hypothesis will be p < p0, p > p0, or p ≠ p0. Step 2: If making a decision, choose a significance level ��. Step 3: Compute the test statistic z = ̂p − p0 √ p0(1 − p0) n . Step 4: Compute the P-value. The P-value is an area under the standard normal curve; it depends on the alternate hypothesis as follows: The P-value is the area to the left of z. z The P-value is the area to the right of z. The P-value is the sum of the areas in the two tails. z −|z| |z| Left-tailed: H1: p < p0 Right-tailed: H1: p > p0 Two-tailed: H1: p ≠ p0 Step 5: Interpret the P-value. If making a decision, reject H0 if the P-value is less than or equal to the significance level ��. Step 6: State a conclusion.
navidi_monk_elementary_statistics_2e_ch7-9
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