Question

5. Use the given data set to complete parts​ (a) through​ (c) below.​ (Use a=0.05.)

x              y

10           7.45

8              6.77

13           12.75

9              7.11

11           7.81

14           8.84

6              6.08

4              5.39

12           8.14

7              6.43

5              5.73

a. Construct a scatterplot

Find the linear correlation​ coefficient, r, then determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.

The linear correlation coefficient is r=_____.

​(Round to three decimal places as​ needed.)

Using the linear correlation coefficient found in the previous​ step, determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables. Choose the correct answer below.

A.There is sufficient evidence to support the claim of a nonlinear correlation between the two variables.

B.There is insufficient evidence to support the claim of a nonlinear correlation between the two variables.

C.There is sufficient evidence to support the claim of a linear correlation between the two variables.

D.There is insufficient evidence to support the claim of a linear correlation between the two variables.

c. Identify the feature of the data that would be missed if part​ (b) was completed without constructing the scatterplot. Choose the correct answer below.

A.The scatterplot reveals a distinct pattern that is a​ straight-line pattern with negative slope.

B.The scatterplot reveals a distinct pattern that is not a​ straight-line pattern.

C.The scatterplot reveals a distinct pattern that is a​ straight-line pattern with positive slope.

D.The scatterplot does not reveal a distinct pattern.

Answer 1

Solution:

The scatter plot for the given data is given as below:

From above scatter plot, it is observed that there is a positive linear relationship exists between the given two variables.

The linear correlation coefficient is given as r = 0.815.

This correlation coefficient indicate that there is a strong positive association or correlation exists between the given two variables x and y.

Here, we have to use t test for population correlation coefficient.

H₀: ρ = 0 versus H_a: ρ ≠ 0

Where, ρ is population correlation coefficient.

We are given r = 0.815, n = 11, df = n – 2 = 11 – 2 = 9, α = 0.05

The test statistic formula is given as below:

t = r*sqrt(n – 2)/sqrt(1 – r^2)

t = 0.815*sqrt(11 - 2)/sqrt(1 - 0.815^2)

t = 4.219439

P-value = 0.0022

(by using t-table)

P-value < α = 0.05

So, we reject the null hypothesis

C. There is sufficient evidence to support the claim of a linear correlation between the two variables.

c. Identify the feature of the data that would be missed if part​ (b) was completed without constructing the scatterplot.

C.The scatterplot reveals a distinct pattern that is a​ straight-line pattern with positive slope.

(See the above scatterplot which indicate a positive linear relationship between two variables.)