# Constructing graphic representation of random samples

Question 1

To compare commuting times in various locations, independent random samples were obtained from the six cities presented in the "Longest Commute to Work" graphic on page 255 in your textbook. The samples were from workers who commute to work during the 8:00 a.m. rush hour. One-way Travel to Work in Minutes

Atlanta Boston Dallas Philadelphia Seattle St. Louis

29 18 42 29 30 15

21 37 25 20 23 24

20 27 26 33 31 42

15 25 32 37 39 23

37 32 20 42 14 33

26 34 26 18

48 35

â€¢ Construct a graphic representation of the data using six side-by-side dot plots.

â€¢ Visually estimate the mean commute time for each city and locate it with an X.

â€¢ Does it appear that different cities have different effects on the average amount of time spent by workers who commute to work during the 8:00 a.m. rush hour? Explain.

â€¢ Does it visually appear that different cities have different effects on the variation in the amount of time spent by workers who commute to work during the 8:00 a.m. rush hour? Explain.

Part 2

â€¢ Calculate the mean commute time for each city depicted.

â€¢ Does there seem to be a difference among the mean one-way commute times for these six cities?

â€¢ Calculate the standard deviation for each city's commute time.

â€¢ Does there seem to be a difference among the standard deviations between the one-way commute times for these six cities?

Part 3

â€¢ Construct the 95% confidence interval for the mean commute time for Atlanta and Boston.

â€¢ Based on the confidence intervals found does it appear that the mean commute time is the same or different for these two cities (Atlanta and Boston). Explain

â€¢ Construct the 95% confidence interval for the mean commute time for Dallas.

â€¢ Based on the confidence intervals found in (Atlanta and Boston) and Dallas does it appear that the mean commute time is the same or different for Boston and Dallas? Explain.

â€¢ Based on the confidence levels found in (Atlanta and Boston) and (Dallas) does it appear that the mean commute time is the same or different for the set of three cities, Atlanta, Boston, and Dallas? Explain

â€¢ How doe your confidence intervals compare to the intervals given for Atlanta, Boston, and Dallas in "Longest Commute to Work" on page 255?

Question 2

Interstate 90 is the longest of the east-west U.S. interstate highways with its 3,112 miles stretching from Boston, MA at I-93 on the eastern end to Seattle WA at the Kingdome on the western end. It travels across 13 northern states; the number of miles and number of intersections in each of those states is listed below.

State No. of Inter Miles

WA 57 298

ID 15 73

MT 83 558

WY 23 207

SD 61 412

MN 52 275

WI 40 188

IL 19 103

IN 21 157

OH 40 244

PA 14 47

NY 48 391

MA 18 159

â€¢ Construct a scatter diagram of the data.

â€¢ Find the equation for the line of best fit using x= miles and y=intersections.

â€¢ Using the equation found in part (b), estimate the average number of intersections per mile along I-90.

â€¢ Find a 95% confidence interval for Î²1.

â€¢ Explain the meaning of the interval found in part d.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

Please see the attachments for the solution to the entire question.

Question 1

To compare commuting times in various locations, independent random samples were obtained from the six cities presented in the "Longest Commute to Work" graphic on page 255 in your textbook. The samples were from workers who commute to work during the 8:00 a.m. rush hour. One-way Travel to Work in Minutes

Atlanta Boston Dallas Philadelphia Seattle St. Louis

29 18 42 29 30 15

21 37 25 20 23 24

20 27 26 33 31 42

15 25 32 37 39 23

37 32 20 42 14 33

26 34 26 18

48 35

â€¢ Construct a graphic representation of the data using six side-by-side dot plots.

Minitab output

â€¢ Does it appear that different cities have different effects on the average amount of time spent by workers who commute to work during the 8:00 a.m. rush hour? Explain.

From the dot plots, it is clear that the observations are more or less evenly distributed in the scale and hence there is no significant difference in the average amount of time spent by workers who commute to work during the 8:00 a.m. rush hour. Hence we can conclude that different cities have no significantly different effects on the average amount of time spent by workers who commute to work during the 8:00 a.m. rush hour

â€¢ Does it visually appear that different cities have different effects on the variation in the amount of time spent by workers who commute to work during the 8:00 a.m. rush hour? Explain.

From the box plots, it is clear that the lower and upper end of the graphs is more or less the same. That is, the range within which the observations are distributed is more or less the same. Thus there is no significant difference in the variation in the amount of time spent by workers who commute to work during the 8:00 a.m. rush hour. Hence we can conclude that different cities have no significantly different effects on the variation in the amount of time spent by workers who commute to work during the 8:00 a.m. rush hour.

Part 2

â€¢ Calculate the mean commute time for each city depicted.

Mean =

Atlanta Boston Dallas Philadelphia Seattle St. Louis

Sample Mean 24.66666667 31.57142857 29.42857143 32.2 27.4 25.83333333

â€¢ Does there seem to be a difference among the mean one-way commute times for these six cities?

There might be significant difference in the mean one-way commute times for these six cities between Atlanta and Boston, Atlanta and Dallas, Atlanta and Philadelphia and Boston and St. Louis.

â€¢ Calculate the standard deviation for each city's commute time.

Sample standard deviation =

Atlanta Boston Dallas Philadelphia Seattle St. Louis

Sample SD 7.763160868 9.606545388 7.390470054 8.348652586 9.396807969 10.02829331

â€¢ Does there seem to be a difference among the standard deviations between the one-way commute times for these six cities?

There is no significant difference among the standard deviations between the one-way commute times for these six cities.

Part 3

â€¢ Construct the 95% confidence interval for the mean commute time for Atlanta and Boston.

Atlanta

The 95% confidence interval is given by , where = 24.66666667, s = 7.763160868, n = 6, = 2.570581835

That is, = (16.52, 32.81)

Thus with 95% confidence we can claim that population mean commute time for Atlanta is within (16.52, 32.81).

Details

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation 7.763160868

Sample Mean 24.66666667

Sample Size 6

Confidence Level 95%

Intermediate Calculations

Standard Error of the Mean 3.169297153

Degrees of Freedom 5

t Value 2.570581835

Interval Half Width 8.146937690

Confidence Interval

Interval Lower Limit 16.52

Interval Upper Limit 32.81

Boston

The 95% confidence interval is given by , where = 31.57142857, s = 9.606545388, n = 7, = 2.446911846

That is, = (22.69, 40.46)

Thus with 95% confidence we can claim that population mean commute time for Boston is within (22.69, 40.46).

Details

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation 9.606545388

Sample Mean 31.57142857

Sample Size 7

Confidence Level 95%

Intermediate Calculations

Standard Error of the Mean 3.630932865

Degrees of Freedom 6

t Value 2.446911846

Interval Half Width 8.884572641

Confidence Interval

Interval Lower Limit 22.69

Interval Upper Limit 40.46

â€¢ Based on the confidence intervals found does it appear that the mean commute time is the same or different for these two cities (Atlanta and Boston). Explain

Since the confidence intervals overlap, it appears that the mean commute time is the same for these two cities (Atlanta and Boston).

â€¢ Construct the 95% confidence interval for the mean commute time for Dallas.

The 95% confidence interval is given by , where = 29.42857143, s = 7.390470054, n = 7, = 2.446911846

That is, = (22.59, 36.26)

Thus with 95% confidence we can claim that population mean commute time for Dallas is within (22.59, 36.26).

Details

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation 7.390470054

Sample Mean 29.42857143

Sample Size 7

Confidence Level 95%

Intermediate Calculations

Standard Error of the Mean 2.793335119

Degrees of Freedom 6

t Value 2.446911846

Interval Half Width 6.835044794

Confidence Interval

Interval Lower Limit 22.59

Interval Upper Limit 36.26

â€¢ Based on the confidence intervals found in (Atlanta and Boston) and Dallas does it appear that the mean commute time is the same or different for Boston and Dallas? Explain.

The confidence intervals found in (Atlanta and Boston) and Dallas overlaps each other. Hence it appears that the mean commute time is the same for Boston and Dallas.

â€¢ Based on the confidence levels found in (Atlanta and Boston) and (Dallas) does it appear that the mean commute time is the same or different for the set of three cities, Atlanta, Boston, and Dallas? Explain

The confidence intervals found in (Atlanta and Boston) and Dallas overlaps each other. Hence it appears that the mean commute time is the same for the set of three cities, Atlanta, Boston, and Dallas.

â€¢ How doe your confidence intervals compare to the intervals given for Atlanta, Boston, and Dallas in "Longest Commute to Work" on page 255?

Please provide the figures.

Question 2

Interstate 90 is the longest of the east-west U.S. interstate highways with its 3,112 miles stretching from Boston, MA at I-93 on the eastern end to Seattle WA at the Kingdome on the western end. It travels across 13 northern states; the number of miles and number of intersections in each of those states is listed below.

State No. of Inter Miles

WA 57 298

ID 15 73

MT 83 558

WY 23 207

SD 61 412

MN 52 275

WI 40 188

IL 19 103

IN 21 157

OH 40 244

PA 14 47

NY 48 391

MA 18 159

â€¢ Construct a scatter diagram of the data.

â€¢ Find the equation for the line of best fit using x= miles and y=intersections.

State No. of Inter, Y Miles, X X^2 Y^2 XY

WA 57 298 88804 3249 16986

ID 15 73 5329 225 1095

MT 83 558 311364 6889 46314

WY 23 207 42849 529 4761

SD 61 412 169744 3721 25132

MN 52 275 75625 2704 14300

WI 40 188 35344 1600 7520

IL 19 103 10609 361 1957

IN 21 157 24649 441 3297

OH 40 244 59536 1600 9760

PA 14 47 2209 196 658

NY 48 391 152881 2304 18768

MA 18 159 25281 324 2862

Total 491 3112 1004224 24143 153410

The general form of simple linear regression is Y= a + bX

Where Y is the dependent variable and X is the independent variable, a and be are known as the regression coefficients .They are estimated by the method of least squares. The estimates of a and b are given by

The parameter b measures the impact of unit change in X on the dependent variable Y. It is the slope of the regression line. The parameter a is the value of Y when X=0. It is known as the Intercept term.

The regression equation can be used to predict the value of Y for a given X. The predicted value of Y is given by

N = 13, = 3112, = 491, = 1004224, = 24143, = 153410

The estimated value of slope b = 0.1384

Estimated intercept a = 4.645

The regression equation, Y = 0.1384 * X + 4.645

That is, Number of intersections = 0.1384 * Miles + 4.645

â€¢ Using the equation found in part (b), estimate the average number of intersections per mile along I-90.

The average number of intersections per mile along I-90 is given by,

Number of intersections = (0.1384 * 3112) + 4.645

= 435.35

â€¢ Find a 95% confidence interval for Î²1.

95% confidence interval for Î²1 is given by,

, where Î²1 = 0.138364119, SE (Î²1) = 0.014920572, = 2.200985159

That is, 0.138364119 Â± 2.200985159 * 0.014920572

= (0.1055, 0.1712)

Details

Coefficients Standard Error Lower 95% Upper 95%

Intercept 4.64698929 4.146952866 -4.480392422 13.774371

Miles 0.138364119 0.014920572 0.105524161 0.171204077

â€¢ Explain the meaning of the interval found in part d.

With 95% confidence we can claim that the population slope is within (0.1055, 0.1712).

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