# 1 Constructing and Interpreting a Confidence Interval

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2 ˆ If our cofidece level is 80% with a sample size of 10: c 80 = ˆ If our cofidece level is 95%, with a sample size of 1000: c 95 =

3 I this problem: c 120 = 1.98 Step 4. Plug everythig ito the formula ad iterpret. The formula for a W% cofidece iterval is: CI W = [ ( ) ( )] s s x c W, x + c W Where c W is foud by lookig at the t-table for 1 degrees of freedom. So for our problem, we ca plug everythig i to get: CI 95% = [ ( ) ( )] , The 95% cofidece iterval is average height of the UCB studet populatio.. This iterval has a 95% chace of coverig the true Practice Use the Stata output below to costruct a 90% cofidece iterval for Michiga State Uiversity udergraduate GPA from a radom sample of the MSU studet body: Variable Obs Mea Std. Dev. Mi Max colgpa Cofidece level: 90% (Sigificace level is =10%) 2. x, s: x = ad s = Fid c 90 : c 90 = Compute & Iterpret iterval: [ ] , [ ] 2.922, Iterpretatio: We are 95% cofidet that this iterval covers the true MSU average GPA. 2 Hypothesis Testig Hypothesis testig is itimately tied to cofidece itervals. Whe we iterpret our cofidece iterval, we specify that there s a 95% chace that the iterval covers the true value. But the there s oly a 5% chace that it does t so after calculatig a 95% cofidece iterval, you would be skeptical to hear that µ is equal to somethig above or below your iterval. To see this, thik about the above practice problem, except suppose (just for the momet) that we kow the ukow true µ ad V ar( x); so we kow that the true MSU average GPA is 3.0 ad V ar( x) = Below is the distributio of x. 3

4 Is there a high probability that a radom sample would yield a x = 2.984? It certaily looks like it, give what we kow about the true distributio of x. Is there a high probability that a radom sample would yield a x = 3.1? Desity mu Hypothesis testig reverses this sceario, but uses the exact same ituitio. Suppose the dea of MSU, Lou Aa Simo, firmly believes the true average GPA of her studets is 3.1, ad she s coviced our radom sample is t a accurate reflectio of the caliber of MSU studets. Suppose we decide to etertai her beliefs for the momet that the true mea GPA is 3.1. Accordig to Lou Aa, the distributio of x looks like this: Desity mu Should we be skeptical of Lou Aa s claim? How to test a hypothesis: Now we ca formalize what we did above ito the 5 steps i hypothesis testig that we covered i lecture: 4

5 Step 1: Defie hypotheses: H 0 ad H 1 Here we write dow the ull ad alterative hypotheses i precise mathematical laguage. H 0 : H 1 : Sice we do t have a good reaso to thik the average GPA should be higher or lower tha 3.1, our alterative hypothesis is: H 1 : µ 3.1 Step 2: Compute test statistic: t = x µ SE( x) = x µ s This step is pretty mechaical. To compute the t-statistic (ofte referred to as the t-stat), we eed the sample mea x, the sample variace s 2, ad the sample size : x = s 2 = = 101 t = = You ca thik of the t-stat as: the sample mea coverted to stadard uits, assumig the ull hypothesis is true. Equivaletly, if the ull hypothesis is true, the we drew the sample mea from a ormal distributio with mea 3.1 ad usig this distributio, we ca re-write x i stadard uits. Step 3: Choose sigificace level (α) ad fid the critical value: c α from table The sigificace level is: Which is the same as the probability of Type 1 Error: Sice Type I error is obviously bad, we wat the probability that we make this type of error to be small. We ca set this probability to be low by choosig a low sigificace level. Oce we ve picked a sigificace level, we ca fid the critical value from a statistical table. Let s choose a 5% sigificace level for our example, so that the probability of Type I Error is just 5%. Check the t-table for the critical value whe our sigificace level for a 2-tailed test is α = 0.05 ad we have 100 degrees of freedom: c α = c.05 = Step 4: Reject the ull hypothesis or fail to reject it: Reject the ull if t > c α If t > c α, the we reject the ull hypothesis. We established above that if the ull hypothesis were true ad the true mea is 3.1, the the probability of observig a sample mea with a t-stat of t > c.05 is equal to the sigificace level, or 5%. So if we do observe a sample mea with a t-stat of t > c.05 the we say we reject the ull hypothesis. Just as before, we ca thik of a picture to help with the ituitio here: 5

8 ˆD = ˆp t ˆp c =.0206 ˆp = (.1447) + (.1653) = SD( ˆD).1526(1.1526).1526(1.1526) = + = z = = Step 3. By the ull hypotheses we chose, we re doig a two-sided test. Let s choose the 5% sigificace level as this is the most commo test that ecoomists evaluate. Check the ormal table to fid that c = Step 4. Reject Fail to reject Step 5. Iterpret:At the 1% sigificace level, there is statistical evidece that the proportio of households with a refrigerator i the cotrol group is ot the same as the proportio of households with a refrigerator i the treatmet group. What does this mea for the study? Probably ot much. I radomized experimets such as this oe, may household characteristics are checked for balace across treatmet ad cotrol. Statistically, we expect that some of our hypothesis tests will reject the ull simply because a 5% sigificace level idicates that 5% of the time we will reject the ull eve though it s true. 8

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