Question

Suppose that (Yį, X;) satisfy the assumptions SLR.1 - SLR.4. A random sample of size n = 250 is drawn and yields Y = 5.4+3.2X, n = 250, R2 = 0.26. (3.1) (1.5) (a) Test Ho: B1 = 0 against H :B1 + 0 at the significance level 5%. (b) Construct a 95% confidence interval for B1. (c) Suppose you learned that Y; and X; were independent. Would you be surprised? Explain. (d) Suppose thet Y; and X, are independent and many samples of size n = 250 are drawn, regression estimated, and (a) and (b) answered. In what fraction of the samples would H, from (a) be rejected? In what fraction of samples would the value B1 = 0 be included in the confidence interval from (b)?

Answer 1

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2 The general fnola f pvalue ix: n-2 > P(tn-a) +P (tn-2a) ap (tn-2>tat) ap Cta50-213) Since naso, teSample &ize is large hus by Centeral lin it Theoen t distibutin knds to nomnal distributim Therefore Pvalue ap (2a-13) 2(0.0166) c 0.0332 The Puabe is leg than o.o5 So/ ue can ject the h pothesi Ho BI-at S-. Signifiane level Construction ot 9. Coniden ce intewal for

3 Using general -fmda tor 951. Confi dence inerval 3.2-1941-),3.2+(6) o.as indupendlert ten: c Suppore y: and i ere Yethe relearcher would be Suyprited it xiond y ae in depen dent woold expect ere independent thun ee Tand y port Ca) States that there Pio Bothe outcome thart P,igo Soggst SYgn ificant statistical euidan ce to ewould be urprired to leanthat xand y ane indipen dent then and i indpendert and naso

24 i given -that iand Y; are indapendent and here are many samples af n= a5d ane drawn oith calulortin Adom portand (h) Compleled SineniSuffici ently lorge and level af sgnifrance expected to be dejected Pn 5-. of Catef tto nd 95of Confidence inteyvals would beantained .