Question

Listed below are annual data for various years. The data are weights​ (metric tons) of imported lemons and car crash fatality rates per​ 100,000 population. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value using α = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between lemon imports and crash fatality​ rates? Do the results suggest that imported lemons cause car​ fatalities?

Lemon Imports	231 231	265 265	358 358	483 483	531 531
Crash Fatality Rate	15.8 15.8	15.6 15.6	15.5 15.5	15.3 15.3	14.8 14.8

What are the null and alternative​ hypotheses?

A.

H0​: ρ=0

H1​: ρ ≠0

B.

H0​: ρ ≠0

H1​: ρ =0

C.

H0​: ρ=0

H1​: ρ <0

D.

H0​: ρ=0

H1​: ρ >0

Construct a scatterplot. Choose the correct graph below.

The linear correlation coefficient r is _____ .

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

The test statistic t is ____.

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

The​ P-value is _______.

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

Because the​ P-value is ______ (greater / less) than the significance level 0.05​, there _______(is not / is) sufficient evidence to support the claim that there is a linear correlation between lemon imports and crash fatality rates for a significance level of α= 0.05.

Do the results suggest that imported lemons cause car​ fatalities?

A.

The results do not suggest any​ cause-effect relationship between the two variables.

B.

The results suggest that an increase in imported lemons causes in an increase in car fatality rates.

C.

The results suggest that imported lemons cause car fatalities.

D.

The results suggest that an increase in imported lemons causes car fatality rates to remain the same.

Answer 1

Here we want to test that : whether there is a linear correlation between lemon imports and crash fatality​ rates.

Therefore null hypothesis ( H₀ ) and alternative hypothesis ( H₁ ) is as follows:

H₀​: ρ = 0

H₁​: ρ ≠ 0

Let's use excel:

Step 1) First enter the given dataset in excel columns

step 2) Select both the column and then click on Insert >>>Scatter >>>First image

so we get the following output:

b) The linear correlation coefficient (r) is obtained by using "=CORREL(range of x, range of y)" this excel command.  

and then enter

So we get r = -0.92749 = -0.927 ( after rounding)

The linear correlation coefficient r is  -0.927

n = total numbe of pairs (x, y) = 5

and level of significance = = 0.05

critical r = -0.92749

The formula of t test statistic is as follows:

Plug the values of n and r in the above formula , we get:

The test statistic t is -4.297

Let's find P-value for two tailed test

P-value = 2* P( t < -4.297 ) = 2* P( t > 4.297 ) = "=TDIST(4.297,3,2)" = 0.023

The​ P-value is 0.023

Because the​ P-value is less than the significance level 0.05​, there is sufficient evidence to support the claim that there is a

linear correlation between lemon imports and crash fatality rates for a significance level of α= 0.05.

Do the results suggest that imported lemons cause car​ fatalities?

The correct option is C

The results suggest that imported lemons cause car fatalities.