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Chapter 8: Hypothesis Testing for Population Proportions We learned in the last chapter how to use confidence intervals to estimate unknown parameters with a given confidence level, that is, within a certain margin of error. Hypothesis testing is a procedure that enables us to choose between two claims regarding an unknown parameter. 8.1: The Essential Ingredients of Hypothesis Testing A statistical hypothesis is a statement about the numerical value of a population parameter. The null hypothesis, H0, represents the neutral, status quo, skeptical claim about a population parameter. It is what we are trying to disprove. The null hypothesis always involves an equality, and it looks like: H0: p = (hypothesized value)
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Sect 8_1.notebook
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The alternative (research) hypothesis, Ha, represents the statement that we are attempting to validate with overwhelming evidence. This hypothesis may involve a negated equality or an inequality. It is a claim that the parameter value is different from the hypothesized value. It looks like: Ha: p ≠ hypothesized value, or Ha: p hypothesized value The alternate hypothesis states that the parameter is greater than, less than, or not equal to the value specified in the null hypothesis. These three cases are defined using the following language:
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Sect 8_1.notebook
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Ex 1: For each of the following, define p and state the null and alternative hypotheses. Is the alternative hypothesis rightsided, leftsided, or twosided? (a) In 2012, the proportion of Americans without health insurance was 15.7%. It is hypothesized that a larger proportion of Americans are uninsured now.
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(b) Hillary Clinton claims the quoted number of 39% percent of the population who approve of Trump’s job performance is an overestimate (Pew Research Center).
(c) Proctor and Gamble claims that Liquid Nyquil contains 10% alcohol. The FDA suspects otherwise, and wishes to conduct a hypothesis test.
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It will be our goal to refute the null hypothesis with overwhelming evidence to some level of confidence (usually 95% or more). If there is sufficient evidence to reject the null hypothesis, then we will accept the alternate hypothesis as being most likely true. If there is not sufficient evidence to reject the null hypothesis, we conclude that the null hypothesis might be true. That is, we will fail to reject it. Caution! We won’t ever conclude that the null hypothesis is true, or make the statement that the null hypothesis is valid.
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Sect 8_1.notebook
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To help make this decision, we choose a significance level, , which is the probability of making the mistake of rejecting the null hypothesis when, in fact, the null hypothesis is true. If we want to be 95% confident in refuting the null hypothesis, we would let . Ex 2: In Example 1(a), suppose . Explain what this value represents in context.
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Sect 8_1.notebook
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The OneProportion ztest Statistic: The evidence we will use in our attempt to refute the null hypothesis is based on a statistic, called the oneproportion z test statistic. If we let p0 represent the value of p that the null hypothesis claims is true, then the idea is to compare our sample proportion to the value p0. The ztest statistic is a z score which measures how far away (in standard deviations) our sample proportion is from the expected value p0.
If is close to p0, then z will be close to zero and we will have no reason to doubt the null hypothesis. The further z is from zero, the more reason we have to doubt the null hypothesis.
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Ex 3: A recent poll indicated that 48% of Americans believe that global warming is mostly due to human activity. A local environmental group believes the percentage is higher. They poll 652 people, and find that 358 believe that humans contribute to global warming. a. What is p? b. Find p0 and .
c. State the null and alternative hypotheses.
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(d) Find the value of the oneproportion ztest statistic.
(e) If the null hypothesis is correct, would you be surprised by this result? Why?
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The sampling distribution for the ztest statistic is often approximately Normal, based on the Central Limit Theorem. Therefore, under certain conditions, we can use the Standard Normal distribution to calculate probabilities regarding the ztest statistic. The probability we are interested in is called a pvalue, which is the probability of getting a result at least as extreme as the result in our hypothesis test, due to random chance alone, assuming that the null hypothesis is true.
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Ex 4: Find the pvalues corresponding with the following hypothesis tests and ztest statistics. Show the area under the N(0, 1) distribution each pvalue represents. (a) H0: p = 0.8, Ha: p
