统计R代写 STAT 300代写 R语言代写

STAT 300 Written Assignment 2

统计R代写 1.(14 marks) The article “Characterization of Highway Runoffff in Austin, Texas, Area” (J. of Envir. Engr., 1998: 131–137) gave a scatter plot,

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1.(14 marks) 统计R代写

The article “Characterization of Highway Runoffff in Austin, Texas, Area” (J. of Envir. Engr., 1998: 131–137) gave a scatter plot, along with the least squares line, of x = rainfall volume (m³ ) and for a y = runoffff volume (m³ ) particular location. The accompanying values were read from the plot.

(a) Create a scatter plot of the data. Does the scatter plot of the data support the use of the simple linear regression model?

(b) Obtain the equation of the least squares line, and interpret its slope.

(d) Calculate a point estimate of the standard deviation σ.

(e) What proportion of observed variation in runoffff volume can be attributed to the simple linear regression relationship between runoffff and rainfall?

(f)Carry out a hypothesis test to decide whether there is a useful linear relationship between rainfall volume and runoffff volume. Use α=0.05

(g) Calculate 95% confifidence interval for the true average change in runoffff volume associated with a 1 m3 increase in rainfall volume

2.(16 marks) 统计R代写

Plywood is made by cutting a thin continuous layer of wood offff a log as it is spun around. Several of these thin layers are then glued together to make plywood sheets. A Forest Products Research Laboratory tested the effffects of certain variables in the production of plywood. Chucks inserted at each end spin the logs, and a saw blade then cuts offff a thin layer. The tests measured the torque of the wood, where torque is the force required to rotate an object about an axis (recorded here in pound–feet).

Thirty logs were used in the fifirst study. A chuck was inserted into each end of each log. The log was turned and a sharp blade was used to cut offff a thin layer of wood. Three temperature levels were used, low (L, 60°F), medium (M, 120°F) and high (H, 150°F). Ten logs were tested at each temperature. The torque that could be applied to the log before the chuck spun out was measured. The data are recorded in the fifile plywood1.txt.

(a)

Create a suitable graphic to display the data. Comment on what you observe.

(b)

Use R to perform ANOVA to test whether it appears that the temperature level has a signifificant effffect on the torque of the wood. Make clear what your test statistic is, the P-value of your test, and your conclusion.

A second study measured the torque that could be applied to the chucks before they spun out, under difffferent conditions of log temperature (in°F), log diameter, and chuck penetration (both in inches). The data are provided in plywood2.txt.

(c)

How many treatments are applied in this experiment?

(d) 统计R代写

Provide a graphical approach to explore which interactions might be present between the factors here. Which pair(s) of factors appear most likely to interact?

(e)

If applying ANOVA on this dataset, why is it not possible to assess the fifit of the “full” model, that is, the model containing all main effffects and interaction terms?

(f)

Using ANOVA on this second dataset, determine which effffect(s), if any, appear to signifificantly impact the torque of the wood. Comment on your results with particular reference to your conclusion in part (b).

(g)

Suppose the torque at the kth level of diameter, the jth level of penetration, and the ith level of temperature is denoted yijk, for i = 1, 2, 3, j = 1, . . . , 4, and k = 1, 2. Write down your proposed model for the data in plywood2.txt, clearly defifining any notation you use.

3.(8 marks)

A researcher has a dataset comprised of scores on three exams for a group of 350 students. The exams, each of which is scored out of 100 points, test mathematical reasoning (MATH), verbal reasoning (VRBL), and general knowledge (GNRL), respectively. The data are described by the following scatterplots.

Not knowing much about statistics, the researcher imports the data into a statistical software package, and starts pushing buttons. He ends up with the three fifitted models whose output is below, but he is not sure what to make of the results.

### FIRST MODEL ###

lm(formula = GNRL ~VRBL + MATH)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.00543 1.83173 3.279 0.00115 **
VRBL 0.62797 0.03822 16.431 < 2e-16 ***
MATH 0.33544 0.04937 6.794 4.75e-11 ***
—
Residual standard error: 8.453 on 347 degrees of freedom
Multiple R-squared: 0.6462, Adjusted R-squared: 0.6441

### SECOND MODEL ###

lm(formula = VRBL ~GNRL + MATH)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.70916 1.90257 4.578 6.56e-06 ***
GNRL 0.69682 0.04241 16.431 < 2e-16 ***
MATH 0.16827 0.05461 3.081 0.00223 **
—
Residual standard error: 8.905 on 347 degrees of freedom
Multiple R-squared: 0.6098, Adjusted R-squared: 0.6075

### THIRD MODEL ###

lm(formula = MATH ~VRBL + GNRL)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 12.21915 1.78310 6.853 3.32e-11 ***
VRBL 0.15825 0.05136 3.081 0.00223 **
GNRL 0.35004 0.05152 6.794 4.75e-11 ***
—
Residual standard error: 8.635 on 347 degrees of freedom
Multiple R-squared: 0.3876, Adjusted R-squared: 0.3841

(a)

Which of the three analyses of the data above address the question of whether MATH and VRBL scores might be associated with each other, amongst students with common GNRL score? Do the data provide evidence that there is such an association, or are they consistent with a lack of association? Justify your answers.

(b)

Give an estimate and 95% confifidence interval for the increase in average MATH score that corresponds to a 20 point increase in VRBL score, amongst students with the same GNRL score. Also, give an estimate and 95% confifidence interval for the increase in average VRBL score that corresponds to a 3.2 point increase in MATH score, amongst students with the same GNRL score.

(c) 统计R代写

It turns out that the researcher also ends up fifitting a fourth model:

### FOURTH MODEL ###

### note than in the R syntax for regression models I() just means treat
### whatever is inside the () as a mathematical expression
lm(formula = MATH ~VRBL + I(VRBL^2) + GNRL + I(GNRL^2))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 44.932002 4.184338 10.738
< 2e-16 ***
VRBL -0.426864 0.199410 -2.141 0.03300 *
I(VRBL^2) 0.006271 0.001973 3.178 0.00162 **
GNRL -0.525516 0.208684 -2.518 0.01225 *
I(GNRL^2) 0.008662 0.002019 4.290 2.33e-05 ***
—
Residual standard error: 7.877 on 345 degrees of freedom
Multiple R-squared: 0.4934, Adjusted R-squared: 0.4875

It also turns out that there is a second group of 100 students writing these exams, and it is reasonable to regard these students as coming from the same population as the fifirst group. In fact, the second group have already written the GNRL and VRBL tests, and the researcher has these scores. They have not yet written the MATH test though. As the result of a silly night in the pub, the researcher is obliged to make a wager with a colleague about which of the 100 students will do worst on the MATH test. Now that he has sobered up, describe how you think the researcher should go about picking which student to bet on?

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