Question

Listed below are amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value using alphaαequals=0.050.05. Is there sufficient evidence to conclude that there is a linear correlation between court incomes and justice​ salaries? Based on the​ results, does it appear that justices might profit by levying larger​ fines? Court Income 66.0 406.0 1567.0 1131.0 271.0 250.0 112.0 153.0 30.0 Justice Salary 31 42 93 58 45 61 25 26 17a. What are the null and alternative​ hypotheses?b. Construct a scatterplot.

Answer 1

Answer:

a) Null hypothesis: $\rho = 0$

Alternative hypothesis: $\rho \neq 0$

b) The scatter plot is on the figure attached.

c) $t=\frac{0.874}{\sqrt{1-(0.874)^2}} \sqrt{9-2}=4.758$  

$p_v = P(t_{7} >4.758) = 0.0010$

Since the p value is lower than the significance level we have enough evidence to reject the null hypothesis at 5% of significance, and we can conclude that the correlation coefficient is significant.

Step-by-step explanation:

Previous concepts

Pearson correlation coefficient(r), "measures a linear dependence between two variables (x and y). Its a parametric correlation test because it depends to the distribution of the data. And other assumption is that the variables x and y needs to follow a normal distribution".

The t distribution (Student’s t-distribution) is a "probability distribution that is used to estimate population parameters when the sample size is small (n<30) or when the population variance is unknown".  

The shape of the t distribution is determined by its degrees of freedom and when the degrees of freedom increase the t distirbution becomes a normal distribution approximately.  

Solution to the problem

In order to calculate the correlation coefficient we can use this formula:  

$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}$  

Let X the Court Income and Y= Justice Salary, for this case we have that:

$n=9, \sum X= 3986, \sumY=398. \sumXY=265160, \sum X^2 =4076636, \sumY^2 =22074$

On this case we got that r =0.874

Part a

Null hypothesis: $\rho = 0$

Alternative hypothesis: $\rho \neq 0$

Part b

The scatter plot is on the figure attached.

Part c

In order to test a hypothesis related to the correlation coefficient we need to use the following statistic:

$t=\frac{r}{\sqrt{1-r^2}} \sqrt{n-2}$

Where n represent the sample size and the statistic t follows a t distribution with n-2 degrees of freedom:

$t \sim t_{n-2}$

On this case our value of n = 9 and the statistic is given by:

$t=\frac{0.874}{\sqrt{1-(0.874)^2}} \sqrt{9-2}=4.758$

And the degrees of freedom are given by df=9-2=7

And the p value for this case since we have a bilateral test is given by:

$p_v = P(t_{7} >4.758) = 0.0010$

Since the p value is lower than the significance level we have enough evidence to reject the null hypothesis at 5% of significance, and we can conclude that the correlation coefficient is significant.