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Section 9.1 ~ Fundamentals of Hypothesis Testing Introduction to Probability and Statistics Ms. Young

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Objective Sec. 9.1 After this section you will understand the goal of hypothesis testing and the basic structure of a hypothesis test, including how to set up the null and alternative hypotheses, how to determine the possible outcomes of a hypothesis test, and how to decide between these possible outcomes.

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Statistical Claims “Of our 350 million users, more than 50% log on to Facebook everyday” “Using Gender Choice could increase a woman’s chance of giving birth to a baby girl up to 80%” “According to the U.S. Census Bureau, Current Population Surveys, March 1998, 1999, and 2000, the average salary of someone with a high school diploma is $30,400 while the average salary of someone with a Bachelor's Degree is $52,200.” How could we determine whether these claims are true or not? –Hypothesis Testing

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Formulating the Hypothesis A hypothesis is a claim about a population parameter Could either be a claim about a population mean, μ, or a population proportion, p All of the claims on the previous slide would be considered hypotheses A hypothesis test is a standard procedure for testing a claim about a population parameter There are always at least two hypotheses in any hypothesis test; the null & alternative hypotheses Sec. 9.1

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Null Hypothesis The null hypothesis, represented as (read as “H-naught”), is the starting assumption for a hypothesis test The null hypothesis always claims a specific value for a population parameter and therefore takes the form of an equality Take the claim, “using Gender Choice could increase a woman’s chance of giving birth to a baby girl up to 80%” for example. If the product did not work, it would be expected that there would be an approximately equally likely chance of having either a boy or a girl. Therefore, the null hypothesis (the claim not working) would be: Sec. 9.1

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Alternative Hypothesis The alternative hypothesis, represented as, is a claim that the population parameter has a value that differs from the value claimed in the null hypothesis, or in other words, the claim does hold true The alternative hypothesis can take one of the following forms: left tailed Ex. ~ A manufacturing company claims that their new hybrid model gets 62 mpg. A consumer group claims that the mean fuel consumption of this vehicle is less than 62 mpg. This alternative hypothesis would be considered left-tailed since the claimed value is smaller (or to the left) of the null value right tailed Ex. ~ The claim that Gender Choice increases a woman’s chance of having a baby girl up to 80% would be testing values above the null value of.5, and would therefore be right-tailed Sec. 9.1

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Alternative Hypothesis Cont’d… two tailed Ex. ~ A wildlife biologist working in the African savanna claims that the actual proportion of female zebras in the region is different from the accepted proportion of 50%. Since the claim does not specify whether the alternative hypothesis is above 50% or below 50%, it would be considered two-tailed in which case the values above and below would be tested Sec. 9.1

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Possible Outcomes of a Hypothesis Test There are two possible outcomes to a hypothesis test: Reject the null hypothesis in which case we have evidence in support of the alternative hypothesis Not reject the null hypothesis in which case we do not have enough evidence to support the alternative hypothesis NOTE – Accepting the null hypothesis is not a possible outcome since it is the starting assumption. The test may provide evidence to NOT REJECT the null hypothesis, but that does not mean that the null hypothesis is true Be sure to formulate the null and alternative hypotheses prior to choosing a sample to avoid bias Sec. 9.1

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Example 1 For the following case, describe the possible outcomes of a hypothesis test and how we would interpret these outcomes The manufacturer of a new model of hybrid car advertises that the mean fuel consumption is equal to 62 mpg on the highway (μ = 62 mpg). A consumer group claims that the mean is less than 62 mpg (μ < 62 mpg). Possible outcomes: Reject the null hypothesis of μ = 62 mpg in which case we have evidence in support of the consumer group’s claim that the mean mpg of the new hybrid is less than 62 Do not reject the null hypothesis, in which case we lack evidence to support the consumer group’s claim Note – this does not necessarily imply that the manufacturer’s claim is true though Sec. 9.1

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Drawing a Conclusion from a Hypothesis Test Using the claim that Gender Choice could increase a woman’s chance of giving birth to a baby girl up to 80%, suppose that a sample produces a sample proportion of,. Although this supports the alternative hypothesis of, is it enough evidence to reject the null hypothesis? This is where statistical significance comes into play (introduced in section 6.1) Recall that something is considered to be statistically significant if it most likely DID NOT occur by chance There are two levels of statistical significance The 0.05 level ~ which means that if the probability of a particular result occurring by chance is less than 0.05, or 5%, then it is considered to be statistically significant at the 0.05 level The 0.01 level ~ which means that if the probability of a particular result occurring by chance is less than 0.01, or 1%, then it is considered to be statistically significant at the 0.01 level The 0.01 level would represent a stronger significance than the 0.05 level Sec. 9.1

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Hypothesis Test Decisions Based on Levels of Statistical Significance We decide the outcome of a hypothesis test by comparing the actual sample result (mean or proportion) to the result expected if the null hypothesis is true (using z-scores). We must choose a significance level for the decision. If the chance that the sample result occurred by chance is less than 0.01, then the test is statistically significant at the 0.01 level and offers STRONG evidence for rejecting the null hypothesis. If the chance that the sample result occurred by chance is less than 0.05, then the test offers MODERATE evidence for rejecting the null hypothesis. If the chance that the sample result occurred by chance is greater than the chosen level of significance (0.01 or 0.05), then we DO NOT reject the null hypothesis. Sec. 9.1

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P-Values A P-Value, or probability value, is the value that represents the probability of selecting a sample at least as extreme as the observed sample In other words, it is the value that allows us to determine if something is statistically significant or not NOTE ~ notice that the P-Value is represented using a capitol P, whereas the population proportion is represented using a lowercase p. We will learn how to actually calculate the P-Value in the following sections A small P-value indicates that the observed result is unlikely (therefore statistically significant) and provides evidence to reject the null hypothesis A large P-value indicates that the sample result is not unusual, therefore not statistically significant - or that it could easily occur by chance, which tells us to NOT reject the null hypothesis Sec. 9.1

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Example 2 You suspect that a coin may have a bias toward landing tails more often than heads, and decide to test this suspicion by tossing the coin 100 times. The result is that you get 40 heads (and 60 tails). A calculation (not shown here) indicates that the probability of getting 40 or fewer heads in 100 tosses with a fair coin is 0.0228. Find the P-value and level of statistical significance for your result. Should you conclude that the coin is biased against heads? The P-Value is 0.0228 This value is smaller than 5% (.05), but not smaller than 1% (.01), so it is statistically significant at the 0.05 level which gives us moderate reason to reject the null hypothesis and conclude that the coin is biased against heads Sec. 9.1

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Putting It All Together Sec. 9.1 Step 1. Formulate the null and alternative hypotheses, each of which must make a claim about a population parameter, such as a population mean (μ) or a population proportion (p); be sure this is done before drawing a sample or collecting data. Based on the form of the alternative hypothesis, decide whether you will need a left-, right-, or two-tailed hypothesis test. Step 2. Draw a sample from the population and measure the sample statistics, including the sample size (n) and the relevant sample statistic, such as the sample mean (x) or sample proportion (p). Step 3. Determine the likelihood of observing a sample statistic (mean or proportion) at least as extreme as the one you found under the assumption that the null hypothesis is true. The precise probability of such an observation is the P-value (probability value) for your sample result. Step 4. Decide whether to reject or not reject the null hypothesis, based on your chosen level of significance (usually 0.05 or 0.01, but other significance levels are sometimes used).

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