Question

In: Statistics and Probability

#72: Note: On this problem, some of your answers might be slightly different from Moodle's because...

#72:

Note: On this problem, some of your answers might be slightly different from Moodle's because of rounding error.

The table below gives the gold medal times for every other Summer Olympics for the women's 100-meter freestyle (swimming).

Year Time (seconds)
1912 82.2
1924 72.4
1932 66.8
1952 66.8
1960 61.2
1968 60.0
1976 55.65
1984 55.92
1992 54.64
2000 53.8
2008 53.1

a. Decide which variable should be the independent variable and which should be the dependent variable.

Independent variable: AnswerYearTime (seconds)

Dependent variable: AnswerYearTime (seconds)

b. Draw a scatter plot of the data.

Please do this on your calculator (or by hand, if you prefer). Take a picture of it, and upload the picture in the appropriate place on Moodle.

c. Does it appear from inspection that there is a relationship between the variables? Why or why not?

Answer

d. Calculate the least squares line. Put the equation in the form of: y = a + bx. Round to two decimals.

y = Answer + Answerx


e. Find the correlation coefficient. Is the decrease in times significant? Round to four decimals.

Correlation coefficient: r = Answer

Is it significant? AnswerYesNo

Explain how you know whether or not it's significant. Your answer must include either a p-value or a critical value and must explain how this value helps you determine whether or not r is significant.

Answer

f. Based on your least squares line, find the estimated gold medal time for 1932. Find the estimated time for 1984.

Round to two decimals.

Year 1932: Estimated gold medal time Answer seconds

What calculation did you do? Answer

Year 1984: Estimated gold medal time Answer seconds

What calculation did you do? Answer

g. Why are the answers from part f different from the chart values?

Answer

h. Does it appear that a line is the best way to fit the data? Why or why not?

AnswerYes, a line is the best way to fit the data.No, a line is not the best way to fit the data.

Explain your reasoning:

AnswerBecause r is significant.Because r is not significant.

i. Use the least-squares line to estimate the gold medal time for the next Summer Olympics (i.e. use the year 2020). Do you think that your answer is reasonable? Why or why not?

Round to two decimals.

Year 2020: Estimated gold medal time Answer seconds

What calculation did you do? Answer

Do you think that your answer is reasonable? Use the textbook's criteria (see Section 12.5 about Prediction).

AnswerYesNo

Explain why or why not, using the textbook's criteria.

Answer

Solutions

Expert Solution

a)
independent - year
dependent -time
b)

c)

yes,

d)

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.951172562
R Square 0.904729242
Adjusted R Square 0.894143603
Standard Error 2.990451442
Observations 11
ANOVA
df SS MS F Significance F
Regression 1 764.3196561 764.3196561 85.46760197 6.85109E-06
Residual 9 80.48519844 8.942799827
Total 10 844.8048545
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 603.4304337 58.5674325 10.30317376 2.78912E-06 470.9416967
Year -0.275602775 0.029811431 -9.244868954 6.85109E-06 -0.343040918

y^ = 603.4304 -0.27560 * year

e)

r = -0.951172562

yes , it is significant

p-value =

6.85109E-06

p-value < 0.05 , hence it is significant

f)

year = 1932

y^ = 603.4304 -0.27560 * year

= 603.4304 -0.27560 * 1932

= 70.9712

for 1984

y^ = 603.4304 -0.27560 *1984

= 56.64

orchestra answered 2 years ago
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