Significance Testing Wed, March 24 th

Statistical Hypothesis Testing wProcedure that allows us to make decisions about pop parameters based on sample stats –Example – mean aver salary of all workers = $28,985 (pop mean =  y) –Sample of 100 African Amer workers, mean salary = $24,100 (samp mean = ybar) –Is that a significant difference from the population? Or not enough of a difference to be meaningful?

Steps of Hypothesis Tests w1) State the Research & Null Hypotheses: –Research Hyp (H1): state what is expected & express in terms of pop parameter –Ex)  y does not = $28,985 (Af Am salary doesn’t = pop salary; the groups differ) –Null Hyp (Ho): usually states there is no difference/effect (opposite of H1). –Ex)  y = $28,985 (Af Am salary = pop salary)

(cont.) –Note: Research Hyp (H1) also known as alternative hypothesis (Ha) wNull hypothesis is tested; we hope to reject it & find support for H1

1 & 2-tailed hypotheses w(Step 1 cont.) – possible to specify 1 or 2- tailed research hyp (H1) –1-tailed is a directional hyp (expect  > some value or < some value) –2-tailed is nondirectional (specify  is not equal to some value) –Use 1-tailed when you can rely on theory to know what to expect; 2-tailed if no prior expectation –Here, we could expect H1:  y < $28,985 (1-tailed test that specifies a lower salary)

Steps 2&3: Select & Calculate the Appropriate Test Statistic wHere, a 1-sample z test to compare a sample mean to a known pop mean –Z = (Ybar –  y ) /  ybar –Where  ybar is std error and =  y / sqrt N and ybar is sample mean;  y is pop mean –Salary example:  y = $28,985,  y = $23,335 ybar = $24,100 and N=100, so… ·  ybar = 23,335 / sqrt(100) = 2,333.5

Example (cont.) –Z = 24,100 – 28,985 / 2,333.5 = wStep 4 – use unit normal table to make a probability decision –Look up z score of –2.09 (or 2.09) in column C (proportion beyond z), find –This is the prob of getting a sample result this extreme ($24,100) if the null hypothesis is true; called p value

Step 4 (cont.): Decision wWe define in advance what is sufficiently improbably to reject the null hypothesis –Find a cutoff point, called  (alpha) below which p must fall to reject null –Usually  =.05,.01, or.001 –Reject null when p <=  –Here, if choose  =.05, we reject null (p =.0183<.05  reject null)

Interpretation –P =.0183 means there is only a 1.83% chance of finding a sample of 100 Afr Amer workers w/mean salary = 24,100 if there is really no difference from overall aver salary. (very unlikely) –Note: if p <  and we reject the null, we can say our findings are ‘statistically significant’; the groups differ significantly

2-tailed test interpretation wOur example used a 1-tailed test, if we’d made a 2-tailed H1 (  y does not = $28,985), we need to adjust the p value –Look up z=-2.09 and find p=.0183, but need to multiple p x 2 if 2-tailed (.0183)x 2 =.0366 –P is still < , so still reject null

Decision Errors wPossible to make 2 types of errors when deciding to reject/fail to reject Ho: Ho is true in reality Ho is false in reality You Reject Ho Type 1 error (  ) Correct You Don’t Reject Ho CorrectType II error

Errors (cont.) wType 1 error = probability of incorrectly rejecting a true Ho wType 2 error = probability of failing to reject a false Ho wSo when  =.05, we have a 5% chance of incorrectly rejecting Ho (Type 1) –Can be more conservative and use  =.01 (1% chance of Type 1), but then increases Type 2 chances…

Lab 17 wSkip #5