Question

The table below lists weights (carats) and prices (dollars) for randomly selected diamonds. Is there sufficient evidence to suggest that there is a linear correlation between weights and prices? Construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r using α = 0.05.

Answer 1

Explanation:

Hello!

Given the variables

X₁: Weight of a diamond (carats)

X₂: Price of a diamond (dollars)

You have to test if there is any correlation between the two variables, the hypotheses are:

H₀: ρ = 0

H₁: ρ ≠ 0

α: 0.05

The resulting correlation coefficient is

r= 0.97

I've used statistics software to calculate the correlation coefficient. To do so manually you have to use the following formula:

`r= (sumX_1X_2-((sumX_1)(sumX_2))/(n) )/([sumX_1^2-((sumX_1)^2)/(n) ][sumX_2^2-((sumX_2)^2)/(n) ])`

The statistic for the parametric test is

`t= (r√(n-2) )/(√((1-r^2)) ) = (0.97√(6-2) )/(√((1-(0.97^2))) ) = 7.98`

p-value 0.0016

The p-value is less than the level of significance, so the decision is to reject the null hypothesis.

Then at a 5% significance level, you can conclude that there is a linear correlation between the weight in carats of the diamonds and their price in dollars.

The scatterplot in the attachment.

I hope you have a SUPER day!