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
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.
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