Statistics – Elements of a Test Hypothesis
Elements of a Test of Hypothesis 1. Null Hypothesis (H0 ) – A statement about the values of population parameters which we accept until proven false. 2. Alternative or Research Hypothesis (Ha )- A statement that contradicts the null hypothesis. It represents researcher’s claim about the population parameters. This will be accepted only when data provides su? cient evidence to establish its truth. 3. Test Statistic – A sample statistic (often a formula) that is used to decide whether to reject H0 . 4. Rejection Region- It consists of all values of the test statistic for which H0 is rejected.
This rejection region is selected in such a way that the probability of rejecting true H0 is equal to ? (a small number usually 0. 05). The value of ? is referred to as the level of signi? cance of the test. 5. Assumptions – Statements about the population(s) being sampled. 6. Calculation of the test statistic and conclusion- Reject H0 if the calculated value of the test statistic falls in the rejection region. Otherwise, do not reject H0 . 7. P-value or signi? cance probability is de? ned as proportion of samples that would be unfavourable to H0 (assuming H0 is true) if the observed sample is considered unfavourable to H0 .
If the p-value is smaller than ? , then reject H0 . Remark: 1. If you ? x ? = 0. 05 for your test, then you are allowed to reject true null hypothesis 5% of the time in repeated application of your test rule. 2. If the p-value of a test is 0. 20 (say) and you reject H0 then, under your test rule, at least 20% of the time you would reject true null hypothesis. 1. Large sample (n > 30) test for H0 : µ = µ0 (known). Z= x ? µ0 ? ? v n Example. A study reported in the Journal of Occupational and Organizational Psychology investigated the relationship of employment status to mental health.
Each of a sample of 49 unemployed men was given a mental health examination using the General Health Questionnaire (GHQ). The GHQ is widely recognized measure of present mental health , with lower values indicating better mental health. The mean and standard deviation of the GHQ scores were x = 10. 94 and s = 5. 10, ? respectively. (a). Specify the appropriate null and alternative hypothesis if we wish to test the research hypothesis that the mean GHQ score for all unemployed men exceeds 10. Is the test one-tailed or two-tailed? (b). If we specify ? = 0. 05, what is the appropriate rejection region for this test? c). Conduct the test, and state your conclusion clearly in the language of this exercise. Find the p-value of the test. (Ans. H0 : µ = 10; Ha : µ > 10; One-tailed test; Rejection region: Z > 1. 645; Test score: Z = 1. 29; Do not reject H0 , GHQ score does not exceeds 10; p-value = 0. 0985) Example. A consumer protection group is concerned that a ketchup manufacturer is ? lling its 20-ounce family-size containers with less than 20 ounces of ketchup. The group purchases 49 family-size bottles of this ketchup, weigh the contents of each, and ? nds that the mean weight is 19. 6 ounces, and the standard deviation is equal to 0. 22 ounces. (a). Do the data provide su? cient evidence for the consumer group to conclude that the mean ? ll per family-size bottle is les than 20 ounces? Test using ? = 0. 05. (b). Find the p-value of the your test in part (a). (Ans. H0 : = 20; Ha : < 20; Rejection Region is Z < ? 1. 645 (one-tailed test); test score Z = ? 4. 45; Reject H0 at ? = 0. 05, su? cient evidence to say that the mean ? ll per family-size bottle is less than 20 ounces; p-value = 0) Example. State University uses thousands of ? uorescent light bulbs each year.
The brand of bulb it currently uses has a mean life of 900 hours. A manufacturer claims that its new brands of bulbs, which cost the same as the brand the university currently uses, has a mean life of more than 900 hours. The university has decided to purchase the new brand if, when tested, the test evidence supports the manufacturer’s claim at the . 10 signi? cance level. Suppose 99 bulbs were tested with the following results: x = 919 hours, s = 86 hours. Find the rejection region for the test of interest to the State University. ? (Ans. Rejection Region: Z > 1. 28) 1 . Small sample (n ? 30) test for H0 : µ = µ0 (known). t= This test requires that the sampled population is normal. x ? µ0 ? s v n Example. A random sample of n observations is selected from a normal population to test the null hypothesis that µ = 10. Specify the rejection region for each of the following combinations of Ha , ? , and n. (a). Ha : µ = 10, ? = 0. 01, n = 14. (Ans. t < ? 3. 012, or t > 3. 012) (b). Ha : µ < 10, ? = 0. 025, n = 26. (Ans. t < ? 2. 06) Example. According to advertisements, a strain of soybeans planted on soil prepared with a speci? d fertilizer treatment has a mean yield of 475 bushels per acre. Twenty farmers who belong to a cooperative plant the soybeans. Each uses a 40-acre plot and records the mean yield per acre. The mean and variance for the sample of 20 farms are x = 462 and s2 = 9070. ? Specify the null and alternative hypothesis used to determine if the mean yield for the soybeans is di? erent than advertised. (Ans. H0 : µ = 475; Ha : µ = 475) Example. A psychologist was interested in knowing whether male heroin addicts’ assessments of self-worth di? er from those of the general male population.
On a test designed to measure assessment of self-worth, the mean score for males from the general population was found to be equal to 48. 6. A random sample of 25 scores achieved by heroin addicts yielded a mean of 44. 1 and a standard deviation of 6. 2. Do the data indicate a di? erence in assessment of self-worth between male heroin addicts and general male population? Test using ? = 0. 01. (Ans. H0 : µ = 48. 6; Ha : µ = 48. 6; Test score t = ? 3. 63 Rejection Region: t > 2. 797, or t < ? 2. 797 (two-tailed test); Observed t-score falls in the rejection region. Reject H0 at ? = 0. 1. Data indicate a di? erence in assessment of self-worth between male heroin addicts and general male population) 3. Large sample test for H0 : p = p0 (known). Z= p ? p0 ? p0 (1? p0 ) n For this test, sample size is considered large if p0 ± 3 p0 (1? p0 ) n falls between 0 and 1. Example. The National Science Foundation, in a survey of 2,237 engineering graduate students who earned their Ph. D. degrees, found that 607 were U. S. citizens; the majority (1,630) of the Ph. D degrees were awarded to foreign nationals. Conduct a test to determine whether the true percentage of engineering Ph.
D. degrees awarded to foreign nationals exceeds 50%. Use ? = 0. 01. (Ans. H0 : p = 0. 5; Ha : p > 0. 5; Test score Z = 21. 63; Rejection region; Z > 2. 33 (one tailed test) Reject H0 at ? = 0. 01. True percentage p exceeds 50%. p-value = 0) Example. The business college computing center wants to determine the proportion of business students who have personal computers (PC’s) at home. If the proportion exceeds 30 percent, then the lab will scale back a proposed enlargement of its facilities. Suppose 250 business students were randomly sampled and 85 have personal computers at home.
Conduct a test to see if the scale back of the proposed enlargement of its facilities is needed. Use ? = 0. 05. (Ans. H0 : p = . 3; Ha : p > 0. 3; Large sample z-test for proportion; test score: Z = 1. 38; Rejection region; Z > 1. 645; Do not reject H0 at ? = 0. 05. Scale back of the proposed enlargement of its facilities is not needed) 2 Example. A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% of the women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately.
A random sample of 70 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in six. Specify the null and alternative hypothesis that the researchers wish to test. Calculate the test statistic, determine the rejection region if ? = 0. 05, ? nd the p-value, and state the conclusion clearly in the language of this exercise. (Ans. H0 : p = 0. 15; Ha : p < 0. 15; Test score: Z = ? 1. 51; Rejection Region: Z < ? 1. 645; Do not reject H0 ; Insu? cient evidence to conclude that the new method is more accurate than the one currently used. -value= p(Z < ? 1. 51) = 0. 5 ? 0. 4345 = 0. 0655) Example. The Midwest Organization of Retired Oncologists and Neurologists (M. O. R. O. N. ) has recently taken ? ack from some of its members regarding the poor choice of the organization’s name. The association bylaws require that more than 60% of the organization must approve a name change. Rather than convene a meeting, it is ? rst desired to use a sample to determine if a meeting is necessary. A random sample of 60 of M. O. R. O. N. ’s members were asked if they want M. O. R. O. N. to change its name. Forty-? ve of the respondent’s said ”yes. Find the p-value for the desired test of hypothesis. (Ans. p-value= p(Z > 2. 37) = 0. 0089) Example. Increasing numbers of businesses are o? ering child-care bene? ts for their workers. However, one union claims that more than 80% of ? rms in the manufacturing sector still do not o? er any child-care bene? ts to their workers. A random sample of 480 manufacturing ? rms is selected, and only 27 of them o? er child-care bene? ts. Specify the rejection region that the union will use when testing at alpha = . 05. (Ans. Ha : p > 0. 8; Rejection region: Z > 1. 645) 3