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45-734 Probability and Statistics II
(4^{th} Mini AY 1997-98 Flex-Mode and Flex-Time)

Assignment #2: Due 31 March 1998

- To assist you in becoming familiar with simple linear regression, it will
be necessary for you to perform the calculations in this problem by hand.
Suppose you are given the following data:

Note: In doing these calculations be careful how much rounding you do! This will become important later. (Keep, say, three significant digits.)**X**_{i}Y_{i}---------- 0 -2 1 -1 2 1 3 1 4 1 - Calculate the k
_{i}values for each observation, that is (as in the Epple Notes),

and demonstrate that:**_ _ k**_{i}= (X_{i}- X_{n})/[å_{i=1,n}(X_{i}- X_{n})^{2}]**å**_{i=1,n}k_{i}= 0, å_{i=1,n}k_{i}x_{i}= 1, and _ å_{i=1,n}k_{i}^{2}= 1/[å_{i=1,n}(x_{i}- X_{n})^{2}] _{^}Calculate the estimated parameters_{^}**b**and_{0}**b**for the linear model. Plot the data by hand as well as your estimated regression equation._{1}- For each observation x
_{i}, obtain point estimates of y_{i}using your estimated regression equation. Demonstrate that_{^}**å**=_{i=1,n}y_{i}**å**_{i=1,n}y_{i} - For each observation, calculate the estimated error
**e**. Show that_{i}**å**= 0 and_{i=1,n}e_{i}

**å**= 0. Calculate the deviation of the observations from their mean:_{i=1,n}e_{i}x_{i}

for each y**_ (y**_{i}- y)_{i}. - Using the values from (d), Calculate the sum of the squared error
(SSE), the total sum of squares (TSS), and the R
^{2}of the regression. _{^}_{^}**s**for this example.^{2}, VAR(b_{0}), and VAR(b_{1})- Perform the hypothesis test:

**H**_{o}: b_{0}= 0

**H**._{1}: b_{0}¹ 0

Do the same test for**b**. Find the 2-tailed P-values for both of these tests. (You can use EVIEWS to do this with the @TDIST(,) command. (See the "Using EVIEWS" handout for an example.)_{1} - Suppose we are in charge of a large manufacturing facility which is heated
by burning a mixture of fuel oil and coal. We have records relating average
hourly temperature and the tonnage of fuel consumed by our heating plant for 8
randomly chosen weeks last year. The data are

These data are in the workfile**Average Hourly Temperature Tons of Fuel Consumed x**_{i}y_{i}---- ---- 28.0 12.4 28.0 11.7 32.5 12.4 39.0 10.8 45.9 9.4 57.8 9.5 58.1 8.0 62.5 7.5

Fuel.wf1.

- Use EVIEWS to estimate the simple linear regression of tons of fuel consumed as a
function of the average hourly temperature.
- Plot the data and the function. How well do the data fit the
equation? Interpret your regression coefficients and hand in the plots.
- Obtain a point estimate of tons of fuel consumed (i.e., the expected
value) for an average hourly temperature of 41 degrees.
- Obtain the 95 percent confidence interval for the point estimate in
(c) and compute the confidence limits.

- Use EVIEWS to estimate the simple linear regression of tons of fuel consumed as a
function of the average hourly temperature.
- Suppose we are in charge of a windmill farm for an electric
utility. We have records relating Wind Velocity (measured in miles per
hour) against direct current output of the windmill. The data are

These data are in the workfile**Wind Velocity DC Output x**_{i}y_{i}---- ----- 2.45 0.123 2.70 0.500 2.90 0.653 3.05 0.558 3.40 1.057 3.60 1.137 3.95 1.144 4.10 1.194 4.60 1.562 5.00 1.582 5.45 1.501 5.80 1.737 6.00 1.822 6.20 1.866 6.35 1.930 7.00 1.800 7.40 2.088 7.85 2.179 8.15 2.166 8.80 2.112 9.10 2.303 9.55 2.294 9.70 2.386 10.00 2.236 10.20 2.310

windmill.wf1.

- Use EVIEWS to estimate the simple linear regression of DC Output
as function of the Wind Velocity.
- Interpret the graph of the residuals. Do the residuals look
random to you?

- Use EVIEWS to estimate the simple linear regression of DC Output
as function of the Wind Velocity.
- Suppose we are in charge of a papermill for a large paper
company. We have records relating of the concentration of hardwoods in
batches of pulp against the the tensile strength (in psi) of the paper
that was manufactured from the pulp. The data are

These data are in the workfile**Hardwood Concentration Tensile Strength x**_{i}y_{i}---- ----- 1.0 6.3 1.5 11.1 2.0 20.0 3.0 24.0 4.0 26.1 4.5 30.0 5.0 33.8 5.5 34.0 6.0 38.1 6.5 39.9 7.0 42.0 8.0 46.1 9.0 53.1 10.0 52.0 11.0 52.5 12.0 48.0 13.0 42.8 14.0 27.8 15.0 21.9

hardwood.wf1.

- Use EVIEWS to estimate the simple linear regression of Tensile Strength
as function of the amount of Hardwood in the pulp.
- Interpret the graph of the residuals. Do the residuals look
random to you?

- Use EVIEWS to estimate the simple linear regression of Tensile Strength
as function of the amount of Hardwood in the pulp.