2 Steps to a Hypothesis Test HypothesesNull Hypothesis (Ho)Alternative Hypothesis (Ha)AlphaDistribution (aka model)Test Statistics and P-valueDecisionConclusion
3 Steps to a Hypothesis Test Can remember the steps by the sentence:“Happy Aunts Make The Darndest Cookies”
4 Example 1– Hypothesis Testing An attorney claims that more than 25% of all lawyers advertise. A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising. At α = 0.05, is there enough evidence to support the attorney’s claim?
5 Hypotheses (Sets up the two sides of the test) Build the Alternative Hypothesis (Ha) first.based on the claim you are testing (you get this from the words in the problem)Three choicesHa: parameter ≠ hypothesized valueHa: parameter < hypothesized valueHa: parameter > hypothesized valueBuild Null Hypothesis (Ho) next.opposite of the Ha (i.e. = , ≥ , ≤ )
6 Example 1– Constructing Hypotheses We need to know what parameter we are testing and which of the three choices for alternative hypothesis we are going to use.“An attorney claims that more than 25% of all lawyers advertise” tells us that this is a test for proportions so our parameter is p.“claims that more than 25%” tells us thatHa: p > .25 and therefore Ho: p ≤ .25
7 Alpha Alpha = α = significance level How much proof we are requiring in order to reject the null hypothesis.The complement of the confidence level that we learned in the last chapterUsually given to you in the problem, if not, you can choose.Most popular alphas: 0.05, 0.01, and 0.10
8 Example 1 – Alpha“At α = 0.05” is given to us in the problem so we just copy α = 0.05
9 ModelThe model is the distribution used for the parameter that you are testing. These are just the same as we used in the confidence intervals.p and μ (n ≥ 30) use the normal distributionμ (n < 30) uses the t-distributionuses the chi-squared distribution
10 Example 1 - ModelThe model used for a proportion is the normal.
11 Test StatisticYou will have a different test statistic for each of the four different parameters that we have learned about.p :μ (n ≥ 30) :
12 Test StatisticYou will have a different test statistic for each of the four different parameters that we have learned about.μ (n < 30) ::
13 p-valueThis is the evidence (probability) that you will get off of your chart and then compare against your criteria (alpha).You will need to find the appropriate probability that goes with your Ha.> and < Ha’s are called one-tailed tests.≠ Ha’s are called two-tailed tests.For z and χ2 you have to take the > probability X2
14 Example 1 – Test Statistic and p-value The formula for a test statistic for proportions is:So, from our problem we need a proportion from a sample (p-hat), the proportion from our hypothesis (po), and a sample size (n).
15 Example 1 – Test Statistic and p-value “A sample of 200 lawyers in a certain city showed that 63 had used some form of advertising” tells us thatp-hat = 63/200 or 0.315From our hypothesis we knowpo = 0.25 (which means that qo = 0.75)“sample of 200” tells us thatn = 200
16 Example 1 – Test Statistic and p-value So our test statistic and p-value are
17 Decision – (always about Ho) We have two choices for decisionReject HoDo Not Reject HoIf our evidence (p-value) is less than α we REJECT Ho.If our evidence (p-value) is greater than α we DO NOT REJECT Ho.
18 Example 1 - Decision Our p-value is 0.0170 and our alpha is 0.05 So, since our p-value is less than our alpha our decision is: REJECT Ho.
19 Conclusion – (always in terms of Ha) ConclusionsReject Ho“There is enough evidence to suggest (Ha).”Do Not Reject“There is not enough evidence to suggest (Ha).”
20 Example 1 - ConclusionOur decision to was to reject Ho, so our conclusion is:“There is enough evidence to suggest that p>0.25”
21 Example 1 - Summary Ho: p ≤ 0.25 Ha: p > 0.25 α = 0.05 Model: Normalz = 2.12 and p-value =Reject HoThere is enough evidence to suggest that p>0.25.
22 Example 2 – Hypothesis Testing A researcher reports that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean of $43,260. At α = 0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the population is $5230.
23 Example 2 (cont.) Hypotheses Alpha Model Ho: μ ≤ $42,000 Ha: μ > $42,000 (given claim is “more than”)Alphaα = 0.05 (given)ModelNormal (n ≥ 30 and it’s a mean)
25 Example 2 (cont.) Decision Conclusion > 0.05 (p-value > alpha)DO NOT REJECT HoConclusionWe do not have evidence to suggest thatμ > $42,000.
26 Example 3 – Hypothesis Testing A physician claims that joggers’ maximal volume oxygen uptake is greater than the average of all adults. A sample of 15 joggers has a mean of 40.6 milliliters per kilogram (ml/kg) and a standard deviation of 6 ml/kg. If the average of all adults is ml/kg, is there enough evidence to support the physicians claim at α = 0.05?
27 Example 3 (cont.) Hypotheses Alpha Model Ho: μ ≤ 36.7 Ha: μ > 36.7 α = 0.05 (given)Modelt(14)
29 Example 3 (cont.) Decision Conclusion (0.01,0.025) < 0.05 (p-value < alpha)REJECT HoConclusionThere is evidence to suggest that μ > 36.7.
30 Example 4 – Hypothesis Testing A researcher knows from past studies that the standard deviation of the time it takes to inspect a car is 16.8 minutes. A sample of 24 cars is selected and inspected. The standard deviation was 12.5 minutes. At α=0.05, can it be concluded that the standard deviation has changed?
31 Example 4 (cont.) Hypotheses Alpha Model Ho: σ = 16.8 Ha: σ ≠ 16.8 α = 0.05 (given)Modelχ2(23)
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