STA4164 Assignment 1 S24 (1)

docx

School

University of West Florida *

*We aren’t endorsed by this school

Course

4164

Subject

Statistics

Date

Feb 20, 2024

Type

docx

Pages

Uploaded by SargentResolve4477

STA4164 – Assignment 1 (100 pts) (Due Feb 12 for section 0001 and Feb 13 for section 0002) [10 pts] Question 1 (Chapter 5): In an analysis of daily soil evaporation (EVAP), Freund (1979) identified the following 10 predictor variables to predict daily soil evaporation:  MAXAT: Maximum daily air temperature  MINAT: Minimum daily air temperature  AVAT: Integrated area under the daily air temperature curve (i.e., a measure of average air temperature)  MAXST: Maximum daily soil temperature  MINST: Minimum daily soil temperature  AVST: Integrated area under the daily soil temperature curve  MAXH: Maximum daily relative humidity  MINH: Minimum daily relative humidity  AVH: Integrated area under the daily relative humidity curve  WIND: Total wind, measured in miles per day In this analysis, maximum daily air temperature was used as the sole predictor of daily soil evaporation. Measurements were recorded across 46 days in June and July. Use the R output below to answer the following questions. Scatterplot: Time vs. Residual Plot:

Residual Plot: Normal Probability Plot:

Call: lm(formula = EVAP ~ MAXAT, data = evap_data) Residuals: Min 1Q Median 3Q Max -32.130 -2.369 1.781 6.195 16.692 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -154.9245 27.3427 -5.666 1.04e-06 *** MAXAT 2.0895 0.3009 6.945 1.38e-08 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 10.22 on 44 degrees of freedom Multiple R-squared: 0.5229, Adjusted R-squared: 0.5121 F-statistic: 48.23 on 1 and 44 DF, p-value: 1.378e-08 (a) Write the least squares estimates of the slope and y-intercept for the regression of daily soil evaporation (Y) on maximum daily air temperature (X). Interpret the slope and y- intercept, if appropriate. Note that daily soil evaporation values range from 1 to 54, and maximum daily air temperatures range from 77 to 97. (b) Conduct a hypothesis test to determine if maximum air temperature is a significant predictor of daily soil evaporation. (c) State the 5 assumptions we are making in this analysis. From the output provided, are any assumptions clearly violated? State why or why not for each assumption. (d) If any assumptions are clearly violated, state a possible fix for the assumption violation.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

[10 pts] Question 2 (5.7): Several research workers associated with the Office of Highway Safety were evaluating the relationship between driving speed (MPH) and the distance a vehicle travels once brakes are applied (DIST). The results of 19 experimental tests are displayed in the following table.

(a) Determine the least-squares estimates of the slope and intercept for each of the following straight-line regressions: Y 1 (DIST) on X and Y 2 ( √ DIST ) on X. (b) Which of the two variable pairs mentioned in part (a) seems to be better suited for straight-line regression? (c) For the variable pair Y 2 and X, test the hypothesis that the true slope is equal to 0 (use α = .01). Be sure to interpret your result. (d) Construct a 99% confidence interval for the true slope in part (c). Interpret your result (you will need to look up a critical t-value for this problem). (e) Estimate the mean value of Y 2 when X = 45. Find a 95% confidence interval for the mean value of Y 2 when X = 45. Interpret your result (Note that observation 4 has X = 45).

[10 pts] Question 3 (6.11): Use the data from question 2. The results of 19 experimental tests are displayed in the following table. For the variable pair Y 2 and X: (a) Determine r and r 2 and interpret the results. (b) Test H 0 : ρ = 0 vs. H a : ρ ≠ 0. Interpret your findings. (c) Find a 95% confidence interval for ρ. Interpret your results regarding the test in part (b). [5 pts] Question 4 (6.2): Examine the five pairs of data points given in the following table. i 1 2 3 4 5 X i -2 -1 0 1 2 Y i 4 1 0 1 4 (a) What is the mathematical relationship between X and Y? (b) Show by computation (formula) that, for the straight-line regression of Y on X, ^ β 1 = 0. (c) Show by computation (formula) that r = 0. (d) Why is there apparently no relationship between X and Y, as indicated by the estimates of β 1 and ρ?

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

[5 pts] Question 5 (7.8): (a) Fill in the blank values in the ANOVA table below. In this analysis, there were 19 observations and only 1 predictor (this is the ANOVA table from questions 2 and 3). Source df Sum of Squares Mean Square F P-Value Regression 399.351 < .0001 Residual 11.180 --------- Total ---------------- --------- (b) Using the table in part (a), perform an F-test for the significance of a straight-line regression.

[20 pts] Question 6 (R Coding Question – Simple Linear Regression): This question involves a dataset of 32 white males over 40 years old from the town of Angina. The information provided for each person includes systolic blood pressure (SBP), body size (QUET), age (AGE), and smoking history (SMK, 0 = nonsmoker, 1 = current or former smoker). This problem will step you through how to conduct a simple linear regression analysis using R, but feel free to use another language if desired (SAS, Python, etc.). Please note that all variable names are pretty general in this example. In practice, you should use descriptive names for your variables. (a) Download the .csv file “HW1Q6.csv” or “HW1Q6.xlsx” from webcourses. Make sure you place the downloaded dataset in whichever file you plan on setting your working directory to in RStudio. Method 1: Open RStudio. Once in RStudio, go to Session -> Set Working Directory -> Choose Directory Choose the folder which has your downloaded data. Use the function to load in the data set into a data frame: dataframe < - read.csv(file = “file_name.csv”)

Method 2: In the top right, click on “Import Dataset,” then “From Excel”. In the top right, click on “Browse” and select the .xlsx file. A preview of the file should appear once done correctly. Click import. To create a dataframe, run the following command: dataframe <- data.frame(Data_Name) For this problem, you can run the following line of code: dataframe <- data.frame(HW1Q6)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Once you do either method 1 or method 2, use the “attach” function to attach the data frame a specify to R what data frame we are using. Simply show the code you used for this part. attach(dataframe) (b) Create a basic scatterplot for age (AGE) vs. blood pressure (SBP) where age is on the x- axis and blood pressure in on the y-axis. Also create a scatterplot of the ln(AGE) vs SBP. The line below will plot a basic scatterplot between var_Y vs var_X. plot(var_X, var_Y) To create a scatterplot between a transformation of X vs. Y, you can apply a function to the variable X within the plot function. This code would plot ln(X) vs. Y: plot(log(var_X), var_Y)) This code would plot X-Squared vs. Y: plot((var_X)^2, var_Y) Comparing your two plots, does it appear that AGE or ln(AGE) has a better linear relationship with SBP? (c) Using your suggested transformation in part (b), estimate the parameters for a simple linear regression model. The code below will create a linear model between the dependent variable “var_Y” and independent variable “var_X”. model <- lm(var_Y ~ var_X, data=dataset) If you decide to transform a variable, you must use the I() function and then the function to transform X. For example, to transform X to ln(X): model <- lm(var_Y ~ I(log(var_X)), data=dataset)

To see the estimates of the parameters: summary(model) Write the least-squares estimates of the y-intercept and slope. (d) To see various diagnostic plots, you can run the following command: plot(model) The first 2 plots created are the most useful. The first plot is a residual plot, and the second plot is a normal probability plot. Does it appear all assumptions are valid? (e) Create an ANOVA table for your model. What is the F-statistic for the F-test for the significance of a straight-line regression? anova(model) (f) Construct a 99% confidence interval for the true slope β 1 . Interpret this interval in context of the problem. Does it appear that age is useful for predicting blood pressure? Explain. confint(model, level=0.99) (g) Predict the blood pressure of a 55-year-old. Construct a 95% confidence interval for the mean blood pressure of a 55-year-old and a 95% prediction interval for a 55-year-old. (Note that if you used ln(AGE), you must do ln(55) in your model rather than 55). predicted_value <- data.frame(AGE=55) #Use if using X predicted_value <- data.frame(SBP=AGE(55)) #Use if using ln(X) predict(model, newdata=predicted_value, interval = 'confidence', level = 0.95) predict(model, newdata=predicted_value, interval = 'prediction', level = 0.95) (h) Between AGE (untransformed) and SBP, find the correlation coefficient, coefficient of determination, and a 95% confidence interval for the correlation. The following code may be useful: cor(var_X, var_Y) cor.test(var_X, var_Y) [10 pts] Question 7 (8.11):

Data on sales revenues Y, television advertising expenditures X 1 , and print media advertising expenditures X 2 for a large retailer for the period 1988–1993 are given in the following table: Part (a): State the model for the regression of sales revenue on television advertising expenditures and print advertising expenditures. Use the accompanying computer output to answer the following questions. Part (b): State the estimate of the model in part (a). Part (c): What is the estimated change in sales revenue for every $1,000 increase in television advertising expenditures (make sure to check the units of X 1 )? Part (d): Find the R 2 -value for the regression of Y on X 1 and X 2 . Interpret your result. (Hint: R 2 = (SSY-SSE)/SSY) Part (e): Predict the sales for a year in which $5 million is spent on TV advertising and $1 million is spent on print advertising. [15 pts] Question 8: In 1990, Business Week magazine compiled financial data on the 1,000 companies that had the biggest impact on the U.S. economy. Data sampled from the top 500

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

companies in Business Week’s report are presented in the following table. In addition to the company name, four variables are shown:  1990 Rank (RANK90) o Based on company’s market value (share price on March 16, 1990, multiplied by available common shares outstanding).  1989 Rank (RANK89) o Rank in 1989 compilation  P-E Ratio (PERATIO) o Price-to-earnings ratio, based on 1989 earnings and March 16, 1990, share price.  Yield (YIELD) o Annual dividend rate as a percentage of March 16, 1990, share price. The dataset is shown here: COMPANY RANK90 RANK89 PERATIO YIELD 1 AT&T 4 4 17 2.87 2 MERCK 7 7 19 2.54 3 BOEING 27 41 24 1.72 4 AMER. HOME PROD. 32 37 14 4.26 5 WALT DISNEY 33 42 23 0.41 6 PFIZER 46 46 14 4.10 7 MCI COMMUNICATIONS 52 72 19 0.00 8 DUNN & BRADSTREET 55 48 15 4.27 9 UNITED TELECOMM. 63 93 22 2.61 10 WARNER LAMBERT 77 91 17 2.94 11 ITT 81 57 8 2.98 12 HUMANA 162 209 15 2.62 13 SALOMON 236 172 7 2.91 14 WALGREEN 242 262 17 1.87 15 LINCOLN NATIONAL 273 274 9 4.73 16 CITIZENS UTILITIES 348 302 21 0.00 17 MNC FINANCIAL 345 398 6 5.46 18 BAUSCH & LOMB 354 391 15 1.99 19 MEDTRONIC 356 471 16 1.10 20 CIRCUIT CITY 497 514 14 0.33 Some R Output is shown below. Let Y=Yield, X 1 = 1990 Rank, X 2 = 1989 Rank, X 3 = P-E Ratio, and X 4 = (P-E ratio) 2 . We want to create the full model: Y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 4 x 4 + E . Find the least-squares estimates of this model.

lm(formula = YIELD ~ RANK90 + RANK89 + PERATIO + I(PERATIO^2), data = df) Residuals: Min 1Q Median 3Q Max -2.2848 -0.7133 0.2326 0.6757 1.2072 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.840196 2.002818 3.915 0.00138 ** RANK90 -0.013758 0.008120 -1.694 0.11084 RANK89 0.008025 0.007370 1.089 0.29338 PERATIO -0.325918 0.269166 -1.211 0.24469 I(PERATIO^2) 0.002146 0.008830 0.243 0.81130 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.096 on 15 degrees of freedom Multiple R-squared: 0.625, Adjusted R-squared: 0.5249 F-statistic: 6.249 on 4 and 15 DF, p-value: 0.003635 > anova(full_model) Analysis of Variance Table Response: YIELD Df Sum Sq Mean Sq F value Pr(>F) RANK90 1 1.8289 1.8289 1.5214 0.2363852 RANK89 1 0.0016 0.0016 0.0013 0.9714506 PERATIO 1 28.1467 28.1467 23.4139 0.0002168 *** I(PERATIO^2) 1 0.0710 0.0710 0.0590 0.8112987 Residuals 15 18.0321 1.2021 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (a) Conduct an overall F-test to determine if at least 1 of the variables significantly helps predict yield. (b) Conduct a partial F-test (or the equivalent t-test) to determine if RANK90 significantly contributes to the prediction of yield, assuming it was added last. (c) Conduct a multiple partial F-test to determine if at least one of P-E ratio and (P-E ratio) 2 contributes significantly to the prediction of yield, assuming the variables are added to a model with RANK90 and RANK89 in it. (d) Conduct a partial F-test to determine if P-E Ratio significantly contributes to the prediction of yield, assuming that P-E Ratio is added to a model with RANK90 and RANK89 in it.

[15 pts] Question 9 (R Coding Question – Multiple Linear Regression) : Using the same data from question 6, answer the following questions. For all hypothesis tests in this section, make sure to include the null and alternate hypothesis, the test statistic, p-value, and conclusion. (a) Consider a model using age (AGE), body size (QUET), and squared body size (QUET 2 ) to predict blood pressure (SBP). What is the least-squares estimates of this model. Make sure you write out the model (not just show R output). full_model <- lm(SBP ~ AGE + QUET + I(QUET^2), data=dataframe summary(full_model) (b) Suppose we want to determine if QUET is useful to predict blood pressure (either QUET or QUET 2 may be useful). To conduct the multiple partial F-test, we can create a reduced model without those terms. reduced_model <- lm(SBP ~ AGE, data=dataframe) We can then use the anova function to conduct the hypothesis test. anova(reduced_model,full_model) Using the output generated above, conduct the hypothesis test to determine if QUET is useful for predicting blood pressure. (c) Using the same method in part (b) or using the output in part (a), conduct the following 2 hypothesis tests: 1. The first for QUET, assuming it was added last to the model. 2. The second for QUET 2 , assuming it was added last to the model. For both tests, state the chosen test (partial F-Test or T-test), the null/alternate hypothesis, the p-value, rejection decision, and conclusion.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version