Question

For 5-6, use the given data to determine if a linear correlation exists. Use proper hypthosesis testing methods. Also determine the regression equation and based on the results of the hypothesis test, find the predicted value. Use a significance level of 0.05.

5) x 1.2 2.7 4.4 6.6 9.5

y 1.6 4.7 9.9 24.5 39.0 Find y if x = 9.

Answer 1

Sample size, n =5

Correlation coefficient, r =0.9865

Null Hypothesis(H0): There is no significant linear relationship between X and Y variables (The correlation coefficient is NOT significantly different from zero).

Alternative Hypothesis(H1): There exists a significant linear relationship between X and Y variables (The correlation coefficient is significantly different from 0). (two-tailed test).

Test statistic, t = =0.9865*/ =10.434

The p-value for t =10.434 and n - 2 =3 degrees of freedom, for a two-tailed test is: p-value =0.0019.

Since the p-value of 0.0019 < 0.05 significance level, we reject the null hypothesis, H0 at 5% significance level.

(By using r-critical value table, at df =3 and at 0.05 significance level, for a two-tailed test, r-critical =±0.878. If r falls within ±0.878, then it is not significantly different from 0 and so, we do not reject H0. If r < -0.878 or if r > 0.878, then we reject H0 and r is significantly different from 0. Since, r =0.9865 > 0.878, it is significantly different from 0 and the null hypothesis, H0 is rejected at 5% significance level).

Thus, there is a sufficient evidence to conclude that there exists a significant linear relationship between X and Y variables because the correlation coefficient is significantly different from 0.

The regression equation is:

ŷ = 4.70332X - 7.01223

Now, at X =9, the predicted Y is:

ŷ = 4.70332(9) - 7.01223 =35.31765