Presentation on theme: "Hypothesis Testing A hypothesis is a claim or statement about a property of a population (in our case, about the mean or a proportion of the population)"— Presentation transcript:
1Hypothesis TestingA hypothesis is a claim or statement about a property of a population (in our case, about the mean or a proportion of the population)A hypothesis test (or test of significance) is a standard procedure for testing a claim or statement about a property of a population.It is extremely important to realize that we are not making definitive conclusions. We are giving probabilistic conclusions. We are either concluding that the results we get are likely due to chance, or unlikely.
2ExamplesIf we flip a coin 100 times, and 52 come up heads, this could easily occur by chance. There is not sufficient evidence to suggest that the coin is unfair.If we flip a coin 100 times, and 75 come up heads, this would be an extremely rare event if the coin was fair. The extremely low probability is evidence that the coin may not be fair.Note: If would be very sloppy of us to conclude in the second example that the coin is definitely unfair. Although extremely rare, 75 heads is still possible by chance from a fair coin.
3Another ExampleA light bulb is advertised as having a mean life of 1000 hours. From a sample, we find the mean life of our sample to be 900 hours. The 95% confidence interval for the population mean is 850 < μ < 1050 hours.We CANNOT conclude:That the actual mean life of light bulbs is 900 hoursThat the advertised life is wrongThat the advertised life is correctWe CAN conclude:From our sample, we are 95% confident that the population mean is between 850 hours and 1050 hours. Since 1000 hours is included in that interval, we do not have sufficient evidence to say that the advertised life is wrong.
4Another approach Claim: The mean life of light bulbs is less than 1000 Working Assumption: The mean life of light bulbs is 1000The sample resulted in a mean life of 900Assuming that μ =1000, the probability that the mean of our sample would be less than 900 is P( < 900) =There are two possible explanations for why our sample came out with a mean life of 900 hours. Either this occurred by chance (with probability 9.5%), or the actual mean life of light bulbs is less than Since the probability (9.5%) isn’t horribly small, we decide that random chance is a reasonable explanation. There isn’t sufficient evidence to support the claim that the mean life of light bulbs is less than 1000 hours.
5Formal Hypothesis Testing The brief process Convert your claim into a symbolic null and alternative hypothesisCalculate a test statisticCompare the test statistic to critical values OR Find a probabilityWrite a conclusion
6Components of a Formal Hypothesis Test The Null hypothesis (denoted H0) is a statement that the value of a population parameter (such as proportion or mean) is equal to some claimed value.The alternative hypothesis (denoted H1 or Ha) is a statement that the value of a population parameter somehow differs from the null hypothesis. The symbolic form must be a >, < or ≠ statement.
7We will be testing the null hypothesis directly (by assuming it’s true) to reach a conclusion to either reject H0 or fail to reject H0.Note: We cannot support a claim that a parameter is equal to a value. So, the null hypothesis must always include equality, and the alternative hypothesis must be inequality.
8ProcessIdentify the claim to be tested and express it in symbolic form.Give the symbolic form that must be true when the original claim is falsePick the one not including equality to be H1, and let the null hypotheses be that the parameter equals the value being considered.
9Example Claim: The mean IQ of statistics students is greater than 110. Symbolic form: μ > 110Opposite: μ ≤ 110H0: μ = 110H1: μ > 110Note: While often your claim will be the alternative hypothesis, it won’t always be.
10Test StatisticsA test statistic is a value computed from the sample data, used in making the decision whether or not to reject the null hypothesis.Z value for proportionZ value for mean (sigma known)T value for mean (sigma unknown)The test statistic indicates how far our sample deviates from the assumed population parameter.
11Critical region and significance Critical region (or rejection region) is the set of all values of the test statistic that cause us to reject the null hypothesis.Significance level (α) is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. Common values are 0.01, 0.05 and 0.10A Critical value is any value that separates the critical region from values of the test statistic that would not cause us to reject the null hypothesis
12ExampleUsing a significance level of α =0.05, lets find the critical value for each of these alternative hypotheses:P ≠ 0.5: Critical region is in two tails of the normal distribution. Using the same method we used in chapter 6, we find the critical values to be z = and z=1.96P < 0.5: The critical region is in the left tail of the normal distribution. Using the methods from 5.2, we find c so P(z < c) = The critical value isP > 0.5: The critical region is in the left tail of the normal distribution. Using the methods from 5.2, we find c so P(z < c) = The critical value is 1.645
13P-ValueThe P-value is the probability of getting a value of the test statistic that is at least as extreme as the one obtained for the sample data. If the P-value is very small (such as less than 0.05), we will reject the null hypothesis.See pullout for help on how to calculate P-value. The exact process depends on your alternative hypothesis.
14Decisions and Conclusions Our final conclusion will always be one of these:Reject the null hypothesisFail to reject the null hypothesisTraditional MethodReject H0 if the test statistic falls within the critical regionOtherwise fail to reject the null hypothesis
15Decisions and Conclusions P-value methodReject H0 if P-value ≤ αFail to reject if H0 > αLess common methodsFind P-value, and leave conclusion to the readerLook at whether population parameter falls in confidence interval estimate
16Final Wording If your original claim contains equality (became H0) Reject H0: “There is sufficient evidence to warrant rejection of the claim that…”Fail to Reject H0: “There is not sufficient evidence to warrant rejection of the claim that…”If your original claim does not contain equality (was H1)Reject H0: “The sample data support the claim that…”Fail to Reject H0: “There is not sufficient sample evidence to support the claim that…”
17Homework7-2: 1-35 every other oddEvery odd recommended.