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  1. Lesson 9: Confidence Intervals and Tests of Hypothesis Two or more samples

  2. The most important part of testing hypothesis • Suppose we are interested in testing whether the population parameter () is equal to k. • H0:  = k • H1:   k • First, we need to get a sample estimate (q) of the population parameter (). • Second, we need to identify the sampling distribution of q, including its mean and variance. • Third, we know in most cases, the test statistics will be in the following form: • t=(q-k)/q • q is the standard deviation of q under the null. The form of q depends on what q is. • Fourth, given the level of significance, determine the rejection region.

  3. Testing a two-sided hypothesis at 5% level of significance Rejection region Rejection region /2 /2 q 0 0+1.96*sq 0 -1.96*sq -1.96 1.96 0 z z=(q- 0)/std(q) is approximately normally distribution under CLT.

  4. The most important part of constructing confidence intervals • Suppose we are interested in constructing a (1-a)*100% confidence interval about the unknown the population parameter (), based on some sampling information. • First, we must have a sample estimate (q) of the population parameter (). • Second, we need to identify the sampling distribution of q, including its mean and variance. • Third, we know in most cases, the following statistics will be approximately normal or student-t distributed: • t=(q-k)/q • q is the standard deviation of q under the null. The form of q depends on what q is. • Fourth, given the confidence level, determine the upper and lower confidence limit for . • q ± ta/2*sq

  5. Constructing a 95% confidence interval for  Upper limit lowerlimit confidence interval /2 /2 q q* q*+1.96*sq q*-1.96*sq -1.96 1.96 0 z z=(q- )/std(q) is approximately normally distribution under CLT. q*: estimate of  from a sample.

  6. Examples of the population parameter of interest • Population mean:  = m • The difference of two population means  = m1– m2 • The sum of two population means  = m1+ m2 • The sum of three population means  = m1+ m2+ m3 • Population variance:  = s2 • Ratio of two population variances:  = s12/s22 Sampling distribution usually normal, due to CLT. Sampling distribution usually chi-square. Sampling distribution usually F.

  7. Distribution of linear combinations of random variables • If m1, m2, and m3 are random variables that are independently normally distributed, • For constants a, b and c, z= am1 + bm2 +cm3 are also normally distributed. • E(z) = aE(m1)+ bE(m2)+cE(m3) • Var(z) = a2Var(m1)+ b2Var(m2)+c2Var(m3)

  8. Distribution of sample variance • Let x1, x2, . . . , xn be a random sample from a population. The sample variance is • The sampling distribution of s2 has mean σ2 • And the following statistics has a 2 distribution with n – 1 degrees of freedom.

  9. Distribution of a ratio of sample variances • The random variable has an F distribution with (nx – 1) numerator degrees of freedom and (ny – 1) denominator degrees of freedom

  10. Two samples Hypothesis testing Constructing confidence interval

  11. An example of hypothesis testing Experimental Group Control Group Placebo Treatment To test the effect of an herbal treatment on improvement of memory you randomly select two samples, one to receive the treatment and one to receive a placebo. Results of a memory test taken one month later are given. Sample 1 Sample 2 The resulting test statistic is 77 - 73 = 4. Is this difference significant or is it due to chance (sampling error)?

  12. Two Sample Tests TEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANS Ho Ho Population 1 Population 1 Population 2 Population 2 H1 H1 Population 1 Population 2 Population 1 Population 2

  13. Comparing two populations • We wish to know whether the distribution of the differences in sample means has a mean of 0. • If both samples contain at least 30 observations we use the z distribution as the test statistic.

  14. Hypothesis Tests for Two Population Means Preferred Format 1 Two-Tailed Test Upper One-Tailed Test Lower One-Tailed Test Format 2

  15. Two Independent Populations: Examples • An economist wishes to determine whether there is a difference in mean family income for households in two socioeconomic groups. • Do HKU students come from families with higher income than CUHK students? • An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools & in urban high schools. • Do students from rural high schools have lower A-level exam score than from urban high schools? Note: The SAT(Scholastic Achievement Test)is a standardized test for college admissions in the United States.

  16. Two Dependent Populations: Examples • An analyst for Educational Testing Service wants to compare the mean GMAT scores of students before & after taking a GMAT review course. • Get HKU graduates to take A-Level English and Chinese exam again. Do they get a higher A-Level English and Chinese exam score than at the time they enter HKU? • Nike wants to see if there is a difference in durability of 2 sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair. Note: The Graduate Management Admissions Test, better known by the acronym GMAT (pronounced G-mat), is a standardized test for determining aptitude to succeed academically in graduate business studies. The GMAT is used as one of the selection criteria by most respected business schools globally, most commonly for admission into an MBA program.

  17. Thinking Challenge Are they independent or dependent? • Miles per gallon ratings of cars before & after mounting radial tires • The life expectancies of light bulbs made in two different factories • Difference in hardness between 2 metals: one contains an alloy, one doesn’t • Tread life of two different motorcycle tires: one on the front, the other on the back dependent independent independent dependent

  18. Comparing two populations • No assumptions about the shape of the populations are required. • The samples are from independent populations. • Values in one sample have no influence on the values in the other sample(s). • Variance formula for independent random variables A and B: V(A-B) = V(A) + V(B) • The formula for computing the value of z is:

  19. EXAMPLE 1 Two cities, Bradford and Kane are separated only by the Conewango River. There is competition between the two cities. The local paper recently reported that the mean household income in Bradford is $38,000 with a standard deviation of $6,000 for a sample of 40 households. The same article reported the mean income in Kane is $35,000 with a standard deviation of $7,000 for a sample of 35 households. At the .01 significance level can we conclude the mean income in Bradford is more?

  20. EXAMPLE 1 continued • Step 1:State the null and alternate hypotheses. H0: µB≤µK ; H1: µB>µK • Step 2: State the level of significance. The .01 significance level is stated in the problem. • Step 3:Find the appropriate test statistic.Because both samples are more than 30, we can use z as the test statistic.

  21. Example 1 continued • Step 4:State the decision rule.The null hypothesis is rejected if z is greater than 2.33. Probability density of z statistic : N(0,1) H0: µB≤ µK ; H1: µB> µK Rejection Region  = 0.01 Acceptance Region  = 0.01

  22. Example 1 continued • Step 5: Compute the value of z and make a decision. H0: µB≤ µK ; H1: µB> µK Rejection Region  = 0.01 Acceptance Region  = 0.01 1.98

  23. Example 1 continued • The decision is to not reject the null hypothesis. We cannot conclude that the mean household income in Bradford is larger.

  24. Example 1 continued • The p-value is: • P(z > 1.98) = .5000 - .4761 = .0239 P-value = 0.0239 H0: µB≤ µK ; H1: µB> µK Rejection Region  = 0.01 1.98

  25. Small Sample Tests of Means • The t distribution is used as the test statistic if one or more of the samples have less than 30 observations. • The required assumptions are: • Both populations must follow the normal distribution. • The populations must have equal standard deviations. • The samples are from independent populations.

  26. Small sample test of means continued • Finding the value of the test statistic requires two steps. Step 1: Pool the sample standard deviations. Why not n1 + n2? Step 2: Determine the value of t from the following formula.

  27. Small sample test of means continued Why not n1 + n2? (n1 – 1) is the degree of freedom. One df is lost because sample mean must be fixed before computation of the sample variance. Division by df instead of n1 ensures the unbiasedness of the s12 as an estimate of the population variance. (n1 +n2 – 2) is the degree of freedom. Two dfs are lost because two sample means must be fixed before computation of the sample variance. Division by df instead of (n1+n2) ensures the unbiasedness of the sp2 as an estimate of the population variance.

  28. EXAMPLE 2 • A recent EPA study compared the highway fuel economy of domestic and imported passenger cars. A sample of 15 domestic cars revealed a mean of 33.7 mpg with a standard deviation of 2.4 mpg. A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9. • At the .05 significance level can the EPA conclude that the mpg is higher on the imported cars?

  29. Example 2 continued • Step 1:State the null and alternate hypotheses. H0: µD≥ µI ; H1: µD< µI • Step 2: State the level of significance. The .05 significance level is stated in the problem. • Step 3:Find the appropriate test statistic.Both samples are less than 30, so we use the t distribution.

  30. EXAMPLE 2 continued Step 4:The decision rule is to reject H0 if t<-1.708. There are 25 degrees of freedom. Probability density of t statistic : t (df=25) Rejection Region  = 0.05

  31. EXAMPLE 2 continued Step 5:We compute the pooled variance:

  32. Example 2 continued We compute the value of t as follows.

  33. Example 2 continued Rejection Region  = 0.05 -1.640 H0 is not rejected. There is insufficient sample evidence to claim a higher mpg on the imported cars.

  34. Hypothesis Testing Involving Paired Observations • Independent samples are samples that are not related in any way. • Dependent samples are samples that are paired or related in some fashion. For example: • If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices. • If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program.

  35. Hypothesis Testing Involving Paired Observations • Use the following test when the samples are dependent: • where is the mean of the differences • is the standard deviation of the differences • n is the number of pairs (differences)

  36. EXAMPLE 3 • An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of eight cities revealed the following information. At the .05 significance level can the testing agency conclude that there is a difference in the rental charged?

  37. EXAMPLE 3 continued

  38. EXAMPLE 3 continued • Step 1: State the null and alternate hypotheses. H0: µd= 0 ; H1: µd≠ 0 • Step 2: State the level of significance. The .05 significance level is stated in the problem. • Step 3: Find the appropriate test statistic. We can use t as the test statistic.

  39. EXAMPLE 3 continued • Step 4:State the decision rule.H0 is rejected if t < -2.365 or t > 2.365. We use the t distribution with 7 degrees of freedom. H0: µd= 0 ; H1: µd≠ 0 Probability density of t statistic : t (df=7) Rejection Region IProbability =0.025 Rejection Region IIprobability=0.025 Acceptance Region  = 0.01

  40. Example 3 continued

  41. Example 3 continued

  42. Example 3 continued • Step 5:Because 0.894 is less than the critical value, do not reject the null hypothesis. There is no difference in the mean amount charged by Hertz and Avis. H0: µd= 0 ; H1: µd≠ 0 0.894 Rejection Region IProbability =0.025 Rejection Region IIprobability=0.025 Acceptance Region  = 0.01

  43. Two Sample Tests of Proportions • We investigate whether two independent samples came from populations with an equal proportion of successes. • The two samples are pooled using the following formula. where X1 and X2 refer to the number of successes in the respective samples of n1 and n2.

  44. Two Sample Tests of Proportions continued • The value of the test statistic is computed from the following formula. Note: The form of standard deviation reflects the assumption of independence of the two samples.

  45. Example 4 • Are unmarried workers more likely to be absent from work than married workers? A sample of 250 married workers showed 22 missed more than 5 days last year, while a sample of 300 unmarried workers showed 35 missed more than five days. Use a .05 significance level.

  46. Example 4 continued • The null and the alternate hypothesis are: H0: U ≤M H1: U > M The null hypothesis is rejected if the computed value of z is greater than 1.65.

  47. Example 4 continued • The pooled proportion is The value of the test statistic is

  48. Example 4 continued • The null hypothesis is not rejected. We cannot conclude that a higher proportion of unmarried workers miss more days in a year than the married workers. • The p-value is: P(z > 1.10) = .5000 - .3643 = .1357

  49. Two Sample Tests TEST FOR EQUAL VARIANCES TEST FOR EQUAL MEANS Ho Ho Population 1 Population 1 Population 2 Population 2 H1 H1 Population 1 Population 2 Population 1 Population 2

  50. Hypothesis Tests of one Population Variance • If the population is normally distributed, Population variance follows a chi-square distribution with (n – 1) degrees of freedom • The test statistic for hypothesis tests about one population variance is Variance under null hypothesis