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Applied Statistics-Homework 8 Solved
Exercise 1: Chick-fil-A Order Type
For this exercise, we’ll consider an extended dataset with nutritional information about menu items from Chick-fil-A. Be sure to use the updated cfa version of the dataset for Homework 8 as posted to Canvas, which is different from the Homework 7 version.
part a
Read in the cfa dataset from Canvas. When you read in the cfa file, include the argument stringsAsFactors = T.
# Use this code chunk for your answer.
setwd("~/Desktop/data")
cfa = read.csv("cfa.csv", stringsAsFactors = T)
part b
What proportion of menu items at Chick-fil-A include chicken?
# Use this code chunk for your answer.
mean(cfa$has_chicken)
## [1] 0.3017241
part c
Fit a model predicting the calories of a menu item from the has_chicken variable. What is the estimate of the difference in mean calories between all menu items that do have chicken and all menu items that do not have chicken?
# Use this code chunk for your answer.
lm1 = lm(Calories ~ has_chicken, data = cfa) summary(lm1)
##
## Call:
## lm(formula = Calories ~ has_chicken, data = cfa)
##
## Residuals:
## Min 1Q Median 3Q Max
## -327.0 -199.8 -103.4 103.0 4660.2
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 259.81 53.32 4.873 3.58e-06 ***
## has_chicken 97.19 97.07 1.001 0.319
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 479.9 on 114 degrees of freedom
## Multiple R-squared: 0.008716, Adjusted R-squared: 2.043e-05
## F-statistic: 1.002 on 1 and 114 DF, p-value: 0.3189
part d
Is there a statistically significant difference in mean calories between all menu items that do have chicken and all menu items that do not have chicken? Explain.
part e
Now, let’s look at the category variable. Create a table that contains a count of how many menu items fall into each possible category. Hint: this can be done with one line of code.
# Use this code chunk for your answer.
table(cfa$category)
##
## breakfast drinks entr\x8ee entree kids salad
## 14 19 2 10 3 3
## sauces side single_item trays treats
## 15 8 19 13 10
part f
What type of variable does R consider or classify the category variable as? If the category variable is included as a first-order term in a linear model, what will its contribution to the p for the model be?
part g
Fit a model predicting the calories of a menu item from the category of that menu item and the serving size. Print a summary of this model.
# Use this code chunk for your answer.
lm2 = lm(Calories ~ category + Serving.size, data = cfa) summary(lm2)
##
## Call:
## lm(formula = Calories ~ category + Serving.size, data = cfa)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1941.95 -73.07 1.27 94.90 2518.77
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 237.85533 94.65898 2.513 0.0135 *
## categorydrinks -561.15767 127.53438 -4.400 2.63e-05 *** ## categoryentr\x8ee 5.73825 264.19282 0.022 0.9827 ## categoryentree 24.44243 144.68634 0.169 0.8662 ## categorykids -152.49420 222.41744 -0.686 0.4945 ## categorysalad 134.24811 223.44177 0.601 0.5493 ## categorysauces -143.08792 130.34197 -1.098 0.2748 ## categoryside -87.16378 154.83768 -0.563 0.5747 ## categorysingle_item -178.69167 123.63256 -1.445 0.1514 ## categorytrays 98.68646 137.84051 0.716 0.4756 ## categorytreats -70.28658 145.18787 -0.484 0.6293
## Serving.size 0.83220 0.09966 8.351 3.13e-13 *** ## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 349.4 on 104 degrees of freedom
## Multiple R-squared: 0.5207, Adjusted R-squared: 0.47
## F-statistic: 10.27 on 11 and 104 DF, p-value: 1.793e-12 part h
What is the baseline level for this model?
part i
From the summary in part g, I notice that one of the estimates is provided as -70.3. What does this value mean?
Answer: We estimate the calories for a menu item of the type treats to have, on average, 70.3 less calories than menu items of the type breakfast, holding constant serving size
Exercise 2: High School Scores
If you haven’t already, you may need to download the faraway package using install.packages(faraway). For our second exercise of Homework 8, we’ll use the hsb dataset included in the faraway package. You can read more about the hsb dataset by using help(hsb)
library(faraway) data(hsb) hsb = hsb
part a
There are 10 variables contained in the High School and Beyond dataset in addition to the id variable, which serves as a record of the observational unit – the student. For each of the 10 variables, record its type, including both the general and specific type.
part b
Fit a model that predicts the math score from the reading score, writing score, high school program, school type, and socioeconomic status. Print the summary, including the coefficients table, of the results. What is the value of p for this model?
# Use this code chunk for your answer.
lm3 = lm(math ~ read + write + prog + schtyp + ses, data = hsb) summary(lm3)
##
## Call:
## lm(formula = math ~ read + write + prog + schtyp + ses, data = hsb)
##
## Residuals:
## Min 1Q Median 3Q Max ## -19.6770 -4.3258 -0.4242 4.4346 17.3644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.72059 3.73370 5.282 3.45e-07 *** ## read 0.35790 0.05811 6.159 4.21e-09 *** ## write 0.29710 0.06179 4.808 3.07e-06 ***
## proggeneral -2.74668 1.21004 -2.270 0.02432 *
## progvocation -3.94757 1.28262 -3.078 0.00239 **
## schtyppublic 0.64310 1.29208 0.498 0.61925 ## seslow -1.53970 1.34073 -1.148 0.25223 ## sesmiddle -0.04555 1.11421 -0.041 0.96743
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.427 on 192 degrees of freedom ## Multiple R-squared: 0.5459, Adjusted R-squared: 0.5294
## F-statistic: 32.98 on 7 and 192 DF, p-value: < 2.2e-16 part c
What is the baseline level for each of the categorical predictors in this model?
part d
Interpret the fitted intercept estimate.
part e
From the output in part b, we’d like to determine if there’s a significant difference in the mean math scores between being from a high socioeconomic class compared to being in a middle socioeconomic class, holding reading scores, writing scores, high school program, and school type constant. What about between students from a high socioeconomic class compared to a low socioeconomic class, holding reading scores, writing scores, high school program, and school type constant? Report your answer to these two tests, including numeric support in your written answer.
# Use this code chunk for your answer, if needed.
summary(lm3)
##
## Call:
## lm(formula = math ~ read + write + prog + schtyp + ses, data = hsb)
##
## Residuals:
## Min 1Q Median 3Q Max ## -19.6770 -4.3258 -0.4242 4.4346 17.3644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.72059 3.73370 5.282 3.45e-07 *** ## read 0.35790 0.05811 6.159 4.21e-09 *** ## write 0.29710 0.06179 4.808 3.07e-06 ***
## proggeneral -2.74668 1.21004 -2.270 0.02432 *
## progvocation -3.94757 1.28262 -3.078 0.00239 **
## schtyppublic 0.64310 1.29208 0.498 0.61925 ## seslow -1.53970 1.34073 -1.148 0.25223 ## sesmiddle -0.04555 1.11421 -0.041 0.96743
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.427 on 192 degrees of freedom ## Multiple R-squared: 0.5459, Adjusted R-squared: 0.5294
## F-statistic: 32.98 on 7 and 192 DF, p-value: < 2.2e-16
part f
We’d like to determine if there’s a statistically significant difference of the mean math scores depending on the high school program, holding reading scores, writing scores, school type, and socioeconomic class constant. We’d like to be able to compare each set of two programs (academic vs. general, academic vs. vocation, & general vs. vocation).
Perform any necessary calculations to determine if there’s a statistically significant difference between each of these sets of two programs. Report your answer for these three tests, including numeric support.
# Use this code chunk for your answer.
lm4 = lm(math ~ prog + read + write + schtyp + ses, data = hsb) summary(lm4)
##
## Call:
## lm(formula = math ~ prog + read + write + schtyp + ses, data = hsb)
##
## Residuals:
## Min 1Q Median 3Q Max ## -19.6770 -4.3258 -0.4242 4.4346 17.3644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 19.72059 3.73370 5.282 3.45e-07 ***
## proggeneral -2.74668 1.21004 -2.270 0.02432 *
## progvocation -3.94757 1.28262 -3.078 0.00239 **
## read 0.35790 0.05811 6.159 4.21e-09 *** ## write 0.29710 0.06179 4.808 3.07e-06 ***
## schtyppublic 0.64310 1.29208 0.498 0.61925 ## seslow -1.53970 1.34073 -1.148 0.25223 ## sesmiddle -0.04555 1.11421 -0.041 0.96743
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.427 on 192 degrees of freedom ## Multiple R-squared: 0.5459, Adjusted R-squared: 0.5294 ## F-statistic: 32.98 on 7 and 192 DF, p-value: < 2.2e-16
hsb$prog = relevel(hsb$prog, ref = 2)
lm7 = lm(math ~ prog + read + write + schtyp + ses, data = hsb) summary(lm7)
##
## Call:
## lm(formula = math ~ prog + read + write + schtyp + ses, data = hsb)
##
## Residuals:
## Min 1Q Median 3Q Max ## -19.6770 -4.3258 -0.4242 4.4346 17.3644
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.97392 3.61889 4.690 5.17e-06 ***
## progacademic 2.74668 1.21004 2.270 0.0243 *
## progvocation -1.20089 1.36388 -0.880 0.3797
## read 0.35790 0.05811 6.159 4.21e-09 *** ## write 0.29710 0.06179 4.808 3.07e-06 ***
## schtyppublic 0.64310 1.29208 0.498 0.6192 ## seslow -1.53970 1.34073 -1.148 0.2522 ## sesmiddle -0.04555 1.11421 -0.041 0.9674
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.427 on 192 degrees of freedom ## Multiple R-squared: 0.5459, Adjusted R-squared: 0.5294
## F-statistic: 32.98 on 7 and 192 DF, p-value: < 2.2e-16
part g
Alicia isn’t sure about including the school type variable and the high school program variable in the model to predict math scores. Alicia would like to perform a single statistical test to decide whether to include these two variables in the model from part b. Help Alicia perform this test. Generate the R output, report the p-value, the decision of the test, and the model that should be used going forward.
# Use this code chunk for your answer.
testing_model = lm(math ~ read + write + ses,
data = hsb)
anova(lm4, testing_model)
## Analysis of Variance Table
##
## Model 1: math ~ prog + read + write + schtyp + ses
## Model 2: math ~ read + write + ses
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 192 7930.5
## 2 195 8374.7 -3 -444.16 3.5844 0.01483 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
part h
Suppose that an additional type of school, a charter school, recently opened in the years since the hsb data were collected. Based on the model from part b, could we calculate a fitted value for a student who attended the charter school? Explain.
Exercise 3: US Wage Model Interpretations
For this exercise, we’ll analyze weekly wages of US male workers in 1988. This data is contained in the uswages dataframe from the faraway package. Before beginning our analyses, the starter code chunk creates a new version of the dataset that is more appropriate for regression purposes.
data(uswages) usawages = uswages
usawages$geo = factor(names(uswages[,6:9])[max.col(uswages[,6:9])]) usawages = usawages[,-c(6:9)] head(usawages)
## wage educ exper race smsa pt geo
## 6085 771.60 18 18 0 1 0 ne ## 23701 617.28 15 20 0 1 0 we ## 16208 957.83 16 9 0 1 0 so ## 2720 617.28 12 24 0 1 0 ne ## 9723 902.18 14 12 0 1 0 mw
## 22239 299.15 12 33 0 1 0 we
For this exercise, we will work with the corrected usawages data (Note the additional “a” in “usa” at the beginning of the data frame).
part a
Fit a model to the usawages data, predicting wage from education, experience, living in a Standard Metropolitan Statistical Area (city + surrounding suburbs), and part time status.
# Use this code chunk for your answer.
lm5 = lm(wage ~ educ + exper + smsa + pt, data = usawages) summary(lm5)
##
## Call:
## lm(formula = wage ~ educ + exper + smsa + pt, data = usawages)
##
## Residuals:
## Min 1Q Median 3Q Max
## -878.6 -213.8 -53.0 126.1 7524.3
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -264.788 50.686 -5.224 1.93e-07 ***
## educ 49.786 3.243 15.354 < 2e-16 *** ## exper 9.075 0.728 12.465 < 2e-16 *** ## smsa 111.825 21.617 5.173 2.54e-07 *** ## pt -340.017 32.027 -10.617 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 413.6 on 1995 degrees of freedom
## Multiple R-squared: 0.1924, Adjusted R-squared: 0.1908 ## F-statistic: 118.9 on 4 and 1995 DF, p-value: < 2.2e-16 part b
There’s a specific vocabulary term that applies to the variable for part time status. What is that vocabulary term?
part c
Interpret the coefficients for education and living in a Standard Metropolitan Statistical Area in part a.
part d
The model from part a could be written out equivalently as 4 distinct models after partitioning the data based on values recorded in 2 variables. Write out each of these 4 models, and define to what part of the data these models apply.
Exercise 4: Summarizing Interaction in US Wages
For this problem, we’ll continue working with the usawages dataset, but this time we’ll focus on a model that includes an interaction term.
part a
Fit a model predicting wage from the geographic area that a male worker lives (geo), the experience level of that worker, and the interaction of the two variables. Print the summary of that model.
# Use this code chunk for your answer.
lm6 = lm(wage ~ geo * exper, data = usawages) summary(lm6)
##
## Call:
## lm(formula = wage ~ geo * exper, data = usawages)
##
## Residuals:
## Min 1Q Median 3Q Max
## -770.6 -274.3 -82.1 165.7 6887.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 448.8918 34.2842 13.093 < 2e-16 *** ## geone 93.5695 49.3678 1.895 0.0582 .
## geoso 14.9600 46.0752 0.325 0.7455 ## geowe 72.5725 51.5719 1.407 0.1595
## exper 7.6816 1.5569 4.934 8.73e-07 ***
## geone:exper -3.0508 2.1506 -1.419 0.1562 ## geoso:exper -1.4928 2.0403 -0.732 0.4645
## geowe:exper -0.3436 2.3835 -0.144 0.8854
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 451.5 on 1992 degrees of freedom
## Multiple R-squared: 0.03909, Adjusted R-squared: 0.03572 ## F-statistic: 11.58 on 7 and 1992 DF, p-value: 1.714e-14 part b
Using the geographic area variable to separate the data into four different partitions, write out the model for each partition.
part c
Visualize the relationship between the experience level of the worker, the geographic area, and the wage. Make sure to include appropriate summary lines in your plot representing the model fitted in part a.
# Use this code chunk for your answer.
ggplot(data = usawages, aes(x = exper, y = wage, color = geo)) +
geom_smooth(method = 'lm', se = F) + geom_point()
## `geom_smooth()` using formula 'y ~ x'
ggplot(data = usawages, aes(x = exper, y = wage, color = geo)) +
geom_smooth(method = 'lm', se = F) # plot w out outliers and points to better see lines
## `geom_smooth()` using formula 'y ~ x'
part d
Perform a single statistical test to test if at least one of the geographic regions has a different slope from the other regions. Report the p-value and a conclusion to the problem, indicating if we have evidence that at least one of the regions has a different slopes. Hint: we are testing for the different geographic regions
simultaneously with one test.
# Use this code chunk for your answer. testing_model2 = lm(wage ~ geo,
data = usawages)
anova(testing_model2)
## Analysis of Variance Table
##
## Response: wage
## Df Sum Sq Mean Sq F value Pr(>F)
## geo 3 1711772 570591 2.7054 0.04395 *
## Residuals 1996 420968877 210906
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
part e
Now, perform a single statistical test to test if at least one of the geographic regions has a different intercept from the other regions, assuming a single, constant slope for experience across all of the geographic regions. Report the p-value and a conclusion to the problem, indicating if we have evidence that at least one of the regions has a different intercept. Hint: we are testing for the different geographic regions simultaneously with
one test.
# Use this code chunk for your answer.
null_model3 = lm(wage ~ geo + exper, data = usawages)
testing_model3 = lm(wage ~ exper, data = usawages)
anova(null_model3, testing_model3)
## Analysis of Variance Table
##
## Model 1: wage ~ geo + exper
## Model 2: wage ~ exper
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1995 406643983
## 2 1998 408494360 -3 -1850377 3.026 0.02851 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Exercise 5: Formatting
The last five points of the assignment will be earned for properly formatting your final document. Check that you have:
• included your name on the document
• properly assigned pages to exercises on Gradescope
• selected page 1 (with your name) and this page for this exercise (Exercise 5)
• all code is printed and readable for each question
• all output is printed
• generated a pdf file
