Question

Please answer all parts, use question #2 to solve #3.

2. For a random sample of size n = 25, the correlation is r = 0.31 for normal random variables X and Y. Answer the questions for the hypothesis test. Use a level of significance of a = 0.08. Ho: p= 0 H1: p0 a. The critical value is Z = b. The test statistic is Z = C. The p-value is d. The hypothesis (should, should not) be rejected. 3. Answer the following question for where Y has been regressed on X1, X2, and X3. Use the linear regression output in the Excel file. Your answers should be rounded to 2 decimal places. a. What is the equation for the line of best fit or regression line? b. The proportion is for the amount of the variability of Y that is explained or accounted for by the model. c. The correlation between the observed values of y and the expected values of y is d. The residual is for when X1,=23, X2 = 220, and X3 = 11. e. What is the expected value of Y when X1.=65 X2 = 200, and X3 = 10?

Answer 1

We would be looking at the first question all parts here:

Q2) a) As we are testing here whether there is a correlation between the two variables, and we are testing it from both sides, therefore this is a two tailed test. For 0.08 level of significance, and n - 2 = 23 degrees of freedom, we have from t distribution tables:
P( t₂₃ < 1.832) = 0.96,

Therefore, P( t₂₃ > 1.832) = 0.04, which means:
P(t₂₃ < -1.832) + P(t₂₃ > 1.832) = 0.04*2 = 0.08

Therefore Z = -1.832, 1.832 are the required critical values here.

b) The test statistic here is computed as:

$t^* = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} = \frac{0.31*\sqrt{23}}{\sqrt{1 - 0.31^2}} = 1.56$

therefore 1.56 is the test statistic value here.

c) As this is a two tailed test, for n - 2 = 23 degrees of freedom, we get the p-value from t distribution tables here:
p = 2P( t₂₃ > 1.56) = 2*0.0662 = 0.1324

therefore 0.1324 is the required p-value here.

d) As the p-value here is 0.1324 > 0.08 which is the level of significance, therefore the test is not significant here, and the null hyopthesis should not be rejected here.