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Question 4 [50 marks] In order to investigate the feasibility of starting a Sun...

Mar 29, 2024

Question 4 [50 marks] In order to investigate the feasibility of starting a Sunday edition for a Daily regional newspaper, information was obtained from a sample of 25 regional newspapers concerning their Daily

(x)

and Sunday

(y)

circulation (in millions of copies). Data are shown in Table 1 below. Assume the significance level is

\alpha=0.05

. 2 \begin{tabular}{lcc} \hline Newspapers & \begin{tabular}{c} Daily \\

(x)

\end{tabular} & \begin{tabular}{c} Sunday \\

(y)

\end{tabular} \\ \hline Baltimore Sun & 0.392 & 0.489 \\ Boston Globe & 0.517 & 0.798 \\ Charlotte Observer & 0.239 & 0.299 \\ Chicago Sun Times & 0.538 & 0.559 \\ Cincinnati Enquirer & 0.199 & 0.349 \\ Denver Post & 0.253 & 0.418 \\ Des Moines Register & 0.206 & 0.345 \\ Hartford Courant & 0.231 & 0.313 \\ Houston Chronicle & 0.450 & 0.621 \\ Kansas City Star & 0.289 & 0.423 \\ Los Angeles Daily News & 0.186 & 0.203 \\ Miami Herald & 0.445 & 0.553 \\ Minneapolis Star Tribune & 0.413 & 0.686 \\ New Orleans Times-Picayune & 0.272 & 0.324 \\ Omaha World Herald & 0.234 & 0.285 \\ Orange County Register & 0.355 & 0.408 \\ Portland Oregonian & 0.338 & 0.441 \\ Providence Journal-Bulletin & 0.197 & 0.268 \\ Rochester Democrat \& Chronicle & 0.238 & 0.257 \\ Rocky Mountain News & 0.374 & 0.433 \\ Sacramento Bee & 0.274 & 0.338 \\ San Francisco Chronicle & 0.570 & 0.704 \\ St. Louis Post-Dispatch & 0.391 & 0.586 \\ St. Paul Pioneer Press & 0.202 & 0.268 \\ Tampa Tribune & 0.322 & 0.308 \\ \hline \end{tabular} Table 1: Daily and Sunday circulation of newspapers. a) i) [5 marks] Calculate

\bar{x}, \bar{y}, S_{x x}, S_{y y}

, and

S_{x y}

. ii) [5 marks] Obtain the estimated simple linear regression line. iii) [5 marks] Is there a significant relationship between Sunday circulation and Daily circulation? Justify your answer by testing whether the slope of the simple linear regression is zero and indicate your conclusion. iv) [5 marks] Obtain the

95 \%

confidence intervals for

\hat{a}

and

\hat{b}

. Justify whether the regression line goes through the origin? b) i) [5 marks] What proportion of the variation in the Sunday circulation is explained by the Daily circulation? ii) [5 marks] Calculate a

95 \%

confidence interval for the correlation coefficient between the two variables. And test the hypothesis that the correlation coefficient between the two variables is zero.

Solution by Steps

step 1

Calculate the mean of the Daily circulation (

\bar{x}

) and the Sunday circulation (

\bar{y}

) by summing the respective values and dividing by the number of observations (n=25)

step 2

Calculate the sample variances

S_{xx}

and

S_{yy}

using the formula:

S_{xx} = \sum (x_i - \bar{x})^2

S_{yy} = \sum (y_i - \bar{y})^2

step 3

Calculate the sample covariance

S_{xy}

using the formula:

S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})

Answer

[Insert calculated values for

\bar{x}, \bar{y}, S_{xx}, S_{yy}, S_{xy}

here]

Key Concept

Mean, variance, and covariance are fundamental statistical measures used to summarize data.

Explanation

The mean gives the average value, variance measures the spread of the data, and covariance indicates the direction of the linear relationship between two variables.

step 4

Obtain the estimated simple linear regression coefficients

\hat{a}

(intercept) and

\hat{b}

(slope) using the formulas:

\hat{b} = \frac{S_{xy}}{S_{xx}}

\hat{a} = \bar{y} - \hat{b}\bar{x}

step 5

Write the estimated simple linear regression line as:

\hat{y} = \hat{a} + \hat{b}x

Answer

[Insert the estimated simple linear regression line here]

Key Concept

Simple linear regression is used to model the relationship between two variables.

Explanation

The regression line represents the best linear fit to the data, minimizing the sum of squared differences between observed and predicted values.

step 6

Test the significance of the relationship between Sunday circulation and Daily circulation by testing the null hypothesis

H_0: \beta = 0

against the alternative hypothesis

H_1: \beta \neq 0

, where

\beta

is the population slope. Use the t-statistic:

t = \frac{\hat{b}}{SE(\hat{b})}

where

SE(\hat{b})

is the standard error of the slope

step 7

Determine the critical t-value from the t-distribution with

n-2

degrees of freedom at the

\alpha = 0.05

significance level. Compare the calculated t-statistic to the critical t-value to decide whether to reject

H_0

Answer

[State whether there is a significant relationship and the conclusion of the hypothesis test]

Key Concept

Hypothesis testing in regression is used to determine if there is a statistically significant relationship between variables.

Explanation

If the calculated t-statistic is greater than the critical t-value, we reject the null hypothesis, indicating a significant relationship.

step 8

Obtain the

95\%

confidence intervals for

\hat{a}

and

\hat{b}

using the formulas:

CI(\hat{a}) = \hat{a} \pm t_{\alpha/2, n-2} \cdot SE(\hat{a})

CI(\hat{b}) = \hat{b} \pm t_{\alpha/2, n-2} \cdot SE(\hat{b})

where

t_{\alpha/2, n-2}

is the critical t-value for

n-2

degrees of freedom and

\alpha/2

step 9

Justify whether the regression line goes through the origin by checking if the confidence interval for

\hat{a}

includes zero

Answer

[Insert the

95\%

confidence intervals for

\hat{a}

and

\hat{b}

, and justify whether the regression line goes through the origin]

Key Concept

Confidence intervals provide a range of values within which we can be confident that the population parameter lies.

Explanation

If the confidence interval for the intercept includes zero, it suggests that the regression line may go through the origin.

step 10

Calculate the coefficient of determination

R^2

to find the proportion of the variation in the Sunday circulation explained by the Daily circulation using the formula:

R^2 = \frac{S_{xy}^2}{S_{xx}S_{yy}}

Answer

[Insert the value of

R^2

here]

Key Concept

The coefficient of determination,

R^2

, measures the proportion of variance in the dependent variable that is predictable from the independent variable.

Explanation

A higher

R^2

value indicates a stronger relationship between the variables.

step 11

Calculate a

95\%

confidence interval for the correlation coefficient

r

between the two variables using Fisher's z-transformation:

z = \frac{1}{2}\ln\left(\frac{1+r}{1-r}\right)

and then find the confidence interval in the z-scale before transforming it back to the r-scale

step 12

Test the hypothesis that the correlation coefficient between the two variables is zero by checking if the confidence interval for

r

includes zero

Answer

[Insert the

95\%

confidence interval for the correlation coefficient and the conclusion of the hypothesis test]

Key Concept

The correlation coefficient measures the strength and direction of the linear relationship between two variables.

Explanation

A confidence interval that does not include zero suggests a significant correlation between the variables.