Question

An agency examined the relationship between the ozone level​ (in parts per million or​ ppm) and the population​ (in millions) of cities. Part of the regression analysis is shown to the right. Complete parts a and b below.

Dependent variable​ is: Ozone

R squaredequals82.2​%

s=5.158 with 16-2=14 df

Variable Coeff SE (Coeff)

Intercept 18.769 2.311

Population 6.701 2.027 ​

a) It is suspected that the greater the population of a​ city, the higher its ozone level. Is the relationship statistically​ significant? Assuming the conditions for inference are​ satisfied, test an appropriate hypothesis and state your conclusion in context. Use a significance level of alphaequals0.05. State the null and alternative hypotheses.

A. Upper H 0​: There is no linear relationship between population and​ ozone, beta 1equals0. Upper H Subscript Upper A​: There is a linear relationship between population and​ ozone, beta 1not equals0.

B. Upper H 0​: There is no linear relationship between population and​ ozone, beta 1equals0. Upper H Subscript Upper A​: There is a positive relationship between population and​ ozone, beta 1greater than0.

C. Upper H 0​: There is a linear relationship between population and​ ozone, beta 1not equals0. Upper H Subscript Upper A​: There is a negative relationship between population and​ ozone, beta 1less than0.

D. Upper H 0​: There is a positive relationship between population and​ ozone, beta 1greater than0. Upper H Subscript Upper A​: There is a negative relationship between population and​ ozone, beta 1less than0.

Determine the​ t-test statistic.

t=____(Round to two decimal places as​ needed.)

Find the​ P-value. ​

P-value=____ ​(Round to four decimal places as​ needed.)

What do you​ conclude?

A. Reject Upper H 0. These data provide evidence of a linear relationship between population and ozone.

B. Reject Upper H 0. These data do not provide evidence of a linear relationship between population and ozone.

C. Fail to reject Upper H 0. These data do not provide evidence of a linear relationship between population and ozone.

D. Fail to reject Upper H 0. These data provide evidence of a linear relationship between population and ozone. ​

b) Do you think that the population of a city is a useful predictor of ozone​ level? Use the values of both Upper R squared and s in your explanation.

A. ​Yes, population accounts for 82.2​% of the variability in the​ residuals, and the residual standard error is 5.158 times less than the ozone mean.

B. ​Yes, population accounts for 82.2​% of the variability in​ ozone, and the residual standard deviation is only 5.158 parts per million.

C. ​No, population accounts for 82.2​% of the variability in the​ residuals, and the residual standard error is 5.158 times greater than the ozone mean.

D. ​No, population accounts for 82.2​% of the variability in​ ozone, and the residual standard deviation is very large ​(5.158 parts per​ million).

Answer 1

Solution:

a) It is suspected that the greater the population of a​ city, the higher its ozone level. Is the relationship statistically​significant? Assuming the conditions for inference are​ satisfied, test an appropriate hypothesis and state your conclusion in context. Use a significance level of alpha equals 0.05. State the null and alternative hypotheses.

Answer:

A. H 0​: There is no linear relationship between population and​ ozone, beta 1 = 0. H A​: There is a linear relationship between population and​ ozone, beta 1 not equals 0.

Determine the​ t-test statistic.

$\small \boldsymbol{t=\frac{6.701}{2.027}=3.31 }$

Find the​ P-value. ​

P-value=0.0052

What do you​ conclude?

A. Reject Upper H 0. These data provide evidence of a linear relationship between population and ozone.

b) Do you think that the population of a city is a useful predictor of the ozone​ level? Use the values of both Upper R squared and s in your explanation.

B. ​Yes, the population accounts for 82.2​% of the variability in​ozone, and the residual standard deviation is only 5.158 parts per million.