Section 8-5 Testing a Claim About a Mean:  Not Known

1. Section 8-5 Testing a Claim About a Mean:  Not Known

2. n = sample size Notation = sample mean m = claimed population mean (from H0) s = sample standard deviation

3. 1) The value of the population standard deviation  is not known. 2) Either or both of these conditions is satisfied: The population is normally distributed or n > 30. Requirements for Testing Claims About a Population Mean (with  Not Known)

4. P-values and Critical Values Found in Table A-3 or by calculator Degrees of freedom (df) = n – 1 Test Statistic for Testing a Claim About a Mean (with  Not Known) x – µ s t= n

5. Example: People have died in boat accidents because an obsolete estimate of the mean weight of men (166.3 lb) was used. A random sample of n = 40 men yielded the mean = 172.55 lb and standard deviation s = 26.33 lb. Do not assume that the population standard deviation  is known. Test the claim that men have a mean weight greater than 166.3 lb.

6. Example: Requirements are satisfied:  is not known, sample size is 40 (n > 30) We can express claim as  > 166.3 lb It does not contain equality, so it is the alternative hypothesis. H0:  = 166.3 lb null hypothesis H1:  > 166.3 lb alternative hypothesis (and original claim)

7. Example: Let us set significance level to  = 0.05 Next we calculate t df = n – 1 = 39 area of 0.05, one-tail yields critical value t = 1.685;

8. Example: = 166.3 or z = 0 Critical value t = 1.685 or t = 1.52 t = 1.501 does not fall in the critical region bounded by t = 1.685, we fail to reject the null hypothesis.

9. Example: Final conclusion: Because we fail to reject the null hypothesis, we conclude that there is not sufficient evidence to support a conclusion that the population mean is greater than 166.3 lb.

10. The critical value in the preceding example was t = 1.782, but if the normal distribution were being used, the critical value would have been z = 1.645.The Student t critical value is larger (farther to the right), showing that with the Student t distribution, the sample evidence must be more extreme before we can consider it to be significant. Normal Distribution Versus Student t Distribution

11. Use software or a TI-83/84 Plus calculator. If technology is not available, use Table A-3 to identify a range of P-values (this will be explained in Section 8.6) P-Value Method

12. Testing hypothesis by TI-83/84 • Press STAT and select TESTS • Scroll down to T-Test press ENTER • Choose Data or Stats. For Stats: • Type in m0: (claimed mean, from H0) • x:(sample mean) • sx:(sample st. deviation) • n:(sample size) • choose H1:m≠m0 <m0 >m0 • (two tails) (left tail) (right tail)

13. (continued) • Press on Calculate • Read the test statistic t=… • and the P-value p=…

14. Section 8-6 Testing a Claim About a Standard Deviation or Variance

15. n= sample size s= sample standard deviation s2 = sample variance  = claimed value of the population standard deviation (from H0 ) 2 = claimed value of the population variance (from H0 ) Notation

16. 1. The sample is a simple random sample. 2. The population has a normal distribution. (This is a much stricter requirement than the requirement of a normal distribution when testing claims about means.) Requirements for Testing Claims About  or  2

17. Chi-Square Distribution Test Statistic

18. Use Table A-4. The degrees of freedom df = n –1. Critical Values for Chi-Square Distribution

19. Table A-4 is based on cumulative areas from the right. Critical values are found in Table A-4by first locating the row corresponding to the appropriate number of degrees of freedom (where df = n –1). Next, the significance level  is used to determine thecorrect column. The following examples are based on a significance level of = 0.05. Table A-4

20. Right-tailed test: needs one critical value Because the area to the right of the critical value is 0.05, locate 0.05 at the top of Table A-4. Critical values

21. Left-tailed test: needs one critical value With a left-tailed area of 0.05, the area to the right of the critical value is 0.95, so locate 0.95 at the top of Table A-4. Critical values

22. Two-tailed test: needs two critical values Critical values are two different positive numbers, both taken from Table A-4 Divide a significance level of 0.05 between the left and right tails, so the areas to the right of the two critical values are 0.975 and 0.025, respectively. Locate 0.975 and 0.025 at top of Table A-4 Critical values

23. Critical values for a two-tailed test

24. Finding a range for P-value • A useful interpretation of the P-value: it is observed level of significance. • Compare your test statistic c2 with critical values shown in Table A-4 on the line with df=n-1 degrees of freedom. • Find the two critical values that enclose your test statistic. Determine the significance levels a1 and a2for those two critical values. • Your P-value is between a1anda2(see examples below)

25. Right-tailed test: If the test statistic c2 is between critical values corresponding to the areas a1 and a2 , then your P-value is between a1 and a2 . Left-tailed test: If the test statistic c2 is between critical values corresponding to the areas 1-a1 and 1-a2 , then your P-value is between a1 and a2 . Examples:

26. Two-tailed test: If the test statistic c2 is between critical values corresponding to the areas a1 and a2 , then your P-value is between 2a1 and 2a2 . Two-tailed test: If the test statistic c2 is between critical values corresponding to the areas 1-a1 and 1-a2 , then your P-value is between 2a1 and 2a2 . (Note: for two-tailed tests, multiply the areas by two) Examples:

27. Finding the exact P-value by TI-83/84 • Use the teststatistic c2 and the calculator function c2 cdfto compute the area of the tail: • c2 cdf(teststat,999,df) gives the area of the right tail • (to the right from the test statistic) • c2 cdf(-999,teststat,df) gives the area of the left tail • (to the left from the test statistic) • Multiply the area of the tail by 2 • if you have a two-tailed test