Question

the following hypothesis are given

H0:p=0

h1:p#0

A random sample of 12 paired observations indicated a correlation of 0.32. Can we

conclude that the correlation in the population is equal to zero? Use 0.05 level of

significance. (3 marks)

the following sample observations were randomly selected

x 4 5 3 6 10

y 4 6 5 7 7

Answer 1

a.

From the data above, the correlation coefficient is $r=0.32$ and $n=12$

The hypothesis tested are,

$H_0:\rho=0\space against \space H_1:\rho\not=0$

and apply the student's t distribution to make a decision.

The test statistic is given as,

$t^*=r\sqrt{n-2}/\sqrt{1-r^2}=0.32\sqrt{10}/\sqrt{1-(0.32)^2}= 1.068092$

$t^*$ is compared with the table value at $\alpha =0.05$ with $n-2=12-2=10$ degrees of freedom.

Now, $t_{\alpha/2,10}=t_{0.05/2,10}=t_{0.025,10}=2.228$ and the null hypothesis is rejected if $|t^*|\gt t_{0.025,10}$

Since $|t^*|=1.068092\lt t_{0.025,10}=2.228$, we fail to reject the null hypothesis and conclude that there is sufficient evidence to show that the correlation coefficient of $r=0.32$  in the population is equal to zero at 5% level of significance.

b.

To find the regression equation, we enter the following commands in $R$.

x=c(4,5,3,6,10)

y=c(4,6,5,7,7)

lm(y~x)

The output for these commands is

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept)      x  

   3.767    0.363 

This shows that the $y-intercept=3.767$ and the $slope=0.363$. The regression line is of the form $y=a*x+b$ where $a$ is the slope and $b$ is the y-intercept. Therefore, the regression line equation is, $y=0.363x+3.767$

When x=7, the value of y is obtained by substituting for x in the regression line equation line as follows,

y=(0.363*7)+3.767= 6.308

Therefore, the value of y when x=7 is 6.308.