1 Tests of Hypotheses about the mean - continued

2 Reminder Two types of hypotheses: H 0 - the null hypothesis (e.g. μ=24) H 1 - the alternative hypothesis (e.g. μ>24) Test statistic: P-value: probability of obtaining values as extreme as or more extreme than the test statistic e.g., P(Z≤-2)= Decision at the α significance level: Reject H 0 if p-value<α

3 Testing hypotheses using a confidence interval: Example: A certain maintenance medication is supposed to contain a mean of 245 ppm of a particular chemical. If the concentration is too low, the medication may not be effective; if it is too high, there may be serious side effects. The manufacturer takes a random sample of 25 portions and finds the mean to be 247 ppm. Assume concentrations to be normal with a standard deviation of 5 ppm. Is there evidence that concentrations differ significantly (α=5%) from the target level of 245 ppm? Hypotheses: H 0 : μ=245 H 1 : μ≠245

4 First, lets examine the Z test statistic: Test statistic: P-value: 2P(Z>2)=2(0.0228)= Decision at 5% significance level: P-value>α  reject H0 The concentration differs from 245

5 Now, examine the hypotheses using a confidence interval α =0.05  confidence level is 1- α = 95% 95% CI: [245.04, ] We are 95% certain that the mean concentration is between and Since 245 is outside this CI - reject H 0. The concentration differs from 245

6 H 0 : μ=μ 0 H 1 : μ≠μ 0 If μ 0 is outside the confidence interval, then we reject the null hypothesis at the α significance level. Note: this method is good for testing two-sided hypotheses only [ confidence interval] Examine the hypotheses using a confidence interval μ0μ0

7 Example Suppose a claim is made that the mean weight μ for a population of male runners is 57.5 kg. A random sample of size 24 yields. [σ is known to be 5 kg]. Based on this, test the following hypotheses: H 0 : μ=57.5 H 1 : μ≠57.5 Answer using: a) A Z test statistic b) A confidence interval

8 a) Test statistic: P-value: 2P(Z>2.45)=2( )=2(.0071)=.0142 Decision at 5% significance level: P-value<α  reject H0 Conclusion: Mean weight differs from 57.5

9 b) α =0.05  confidence level is 1- α = 95% 95% CI: [58, 62] 57.5 is outside this CI - reject H 0. Mean weight differs from 57.5 question?: Would you reject H 0 : μ=59 vs. H 1 : μ≠59? No, because 59 is in the interval [58, 62]

10 Testing hypotheses using Minitab In a certain university, the average grade in statistics courses is 80, and σ=11. A teacher at that university wanted to examine whether her students received higher grades than the rest of the stat classes. She took a sample of 30 students and recorded their grades hypotheses: H 0 :μ=80 H 1 :μ>80 data are: mean:

11 Test statistic: P-value: P(Z>2.51)= =0.006 Decision at 5% significance level: P-value<α  reject H0 conclusion: The grades are higher than 80

12 Minitab….\test of hypotheses.MPJ

13 Column of scores:

14 Choose: Stat > Basic Statistics >1-Sample Z Pick options

15 In the “options” window pick the alternative hypothesis

16 One-Sample Z: scores Test of mu = 80 vs mu > 80 The assumed sigma = 11 Variable N Mean StDev SE Mean scores Variable 95.0% Lower Bound Z P scores In the session window: Test statistic: Z = 2.51 P-value: Decision: reject H0 at the 5% significance level conclusion: The grades are higher than 80

17 Use Minitab to build a confidence interval Choose: Stat > Basic Statistics >1-Sample Z Pick options

18 In the “options” window pick the two-sided alternative hypothesis Pick “not equal” Because a confidence interval is like a two sided hypothesis

19 In the session window: One-Sample Z: scores Test of mu = 80 vs mu not = 80 The assumed sigma = 11 Variable N Mean StDev SE Mean scores Variable 95.0% CI Z P scores ( 81.10, 88.97) A 95% CI: [81.10, 88.97] Equivalent to testing hypotheses: H0:μ=80 H1:μ≠80

20 Questions 1. Suppose H 0 was rejected at α=.05. Answer the following questions as: “Yes”, “No”, “Cannot tell”: (a)Would H 0 also be rejected at α=.03? (b)Would H 0 also be rejected at α=.08? (c)Is the p-value larger than.05?

21 2. Suppose H 0 was not rejected at α=.05. Answer the following questions as: “Yes”, “No”, “Cannot tell”: (a)Would H 0 be rejected at α=.03? (b)Would H 0 be rejected at α=.08? (c)Is the p-value larger than.05?

22 3. A 95% confidence interval for the mean time (in hours) to complete an audit task is: [7.04,7.76]. Use the relation between confidence intervals and two-sided tests to examine the following sets of hypotheses: (a) H 0 : μ=7.5H 1 : μ≠7.5 (α=.05) (b) H 0 : μ=7 H 1 : μ≠7 (α=.05)

23 4. A 90% confidence interval for the mean is: [20.1, 23.5]. We can use the relation between confidence intervals and two-sided tests to examine hypotheses about the mean At what level of significance, α, can we test these hypotheses based on the confidence interval? (a)α=.01 (b)α=.025 (c)α=.05 (d)α=.1

24 5. A 90% confidence interval for the mean is: [20.1, 23.5] has been used for testing the following hypotheses: H 0 : μ=19H 1 : μ≠19 H 0 is rejected at 10% significance level (19 is outside the CI) At what level of significance, α, can we still reject H 0 : μ=19? Answer: We can reject H 0 for α<0.1