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Tutor profile: Maria C.

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Maria C.
College Math and Data Science Tutor
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Questions

Subject: R Programming

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Question:

Create a multiple linear regression model and test the model assumptions hold in R

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Maria C.
Answer:

Step 1: Clean your data. This step usually takes longer than you think. It could include: dealing with unknown values, merging data sources, aligning date ranges, renaming columns, and adjusting data types. The goal is to have one clean dataset with consistent data. Example: renaming column 1 in a data set named cars colnames(cars)[1]<-"Year" # results in column 1 being named "Year" Step 2: Set up multiple linear regression model # define fit fit = lm( (dependent variable) ~ (independentvariable) + (independentvariable) + ... (independent variable) ,data = (clean data set) # prints results. There is a lot more to understanding the results but I won't cover that here summary(fit) Step 3: test the linear regression model assumptions hold. There are two main assumptions to test: normality of the residuals and homoscedasticity. Perform the Shapiro-Wilk test to verify the normality of the residuals. shapiro.test(fit$residuals) # access the residuals stored in the regression fit from earlier # if the P-value is less than 0.05, it is sufficient to reject the null hypothesis One way to test homoscedasticity is with the Breusch-Pagan test bptest(fit) # run the test with the fit from earlier # if the P-value is less than 0.05, it is sufficient to reject the null hypothesis

Subject: Calculus

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Question:

Finding the area between two curves. Find the area below $$f(x)=-x^2+10x-24$$ and above the $$x$$-axis.

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Maria C.
Answer:

Start by sketching the region in which we want to find the area. We want to find the area meaning we will want to set up an integral and evaluate it. First, we need to find the limits on $$x$$ since we were not given any limits. In this case, the limits on $$x$$ are when the parabola $$y=-x^2+10x-24$$ crosses the $$x$$-axis. To find the limit on $$x$$, set the parabola equal to $$0$$. In other words, the limit on $$x$$ occurs when the parabola crossed the line $$y=0$$. Thus $$y=-x^2+10x-24$$ becomes $$0=-x^2+10x-24$$. Factor to find where $$x=0$$. $$-x^2+10x-24=0 \rightarrow -(x-6)(x-4)=0$$ Set $$(x-6)$$ and $$(x-4)$$ equal to zero. $$(x-6)=0 \rightarrow x=6$$ $$(x-4)=0 \rightarrow x=4$$ Thus the limit on $$x$$ is $$[4,6]$$. Now to set up the integral. $$area=\int_4^6( -x^2+10x-24) dx$$. Evaluate the integral by first separating each term. $$\int_4^6( -x^2+10x-24) dx = \int_4^6-x^2dx + \int_4^610x dx + \int_4^6-24dx $$ Then use the power rule on each term. $$\int_4^6-x^2dx + \int_4^610x dx + \int_4^6-24dx = (-\frac{1}{3}x^3+5x^2-24x)\vert_{4}^{6}$$ To finish, recall the Second Fundamental Theorem of Calculus. $$\int_a^b f(x)dx=F(b)−F(a)$$. Thus, $$area= (-\frac{1}{3}x^3+5x^2-24x)\vert_{4}^{6} = (-\frac{1}{3}(6)^3+5(6)^2-24(6))- (-\frac{1}{3}(4)^3+5(4)^2-24(4))$$ $$area=\frac{3}{4}$$ We have found the area between the curves.

Subject: Algebra

TutorMe
Question:

Jon is Sara's younger brother. Jon is 5 years younger than Sara. Write Jon's age with respect to Sara's age.

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Maria C.
Answer:

What are our unknowns? Sara's age and Jon's age. Let Sara's age be $$x$$ and let Jon's age be $$y$$ Jon's age is 5 years younger than Sara's or Jon's age is 5 less than Sara's Thus if Sara's age is $$x$$ and Jon's is 5 less than Sara, Jon's age is $$y = x-5$$

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