Lab Homework:#Use the read.table function to load the data from lab8hw.txt and store as object named hw#Submit all plots#1.1) Create a scatter plot of iq on y axis and score on X axis. How do the variables appear to be related?#1.2) Conduct a linear regression of iq and score#1.3) Do you reject or fail to reject the null hypothesis about the slope? Why? #1.4) What is the interpretation of the coefficient for the slope in #1.3? #1.5) Calculate the correlation coefficient for iq and score#1.6) Calculate the R-squared from the correlation coefficient. What is the interpretation for this R-squared?#1.7) Add the regression line to the plot created in #1.1#1.8) Based on what you see in #1.7, do you have any concerns about the results? Why or why not? #1.9) Create a dataset hm_iq_score that is a new version of hw but without outliers in iq & score columns. Use the command out. Also, create a regression line for this new data set.#1.10) Plot again the data in 1.1, add the regression lines found in 1.7 and 1.9 (use different colors to plot those lines). Explain why the regression lines look either very similar or very different. -------------------------------------------------------------------------------------------------------------------------Lab lecture notes from class for your reference: #Lab 8-Contents#1. Scatter Plots in R#2. Linear Regression in R#3. Outliers in Regression#4. Hypothesis testing in Regression#5. Correlation and R-Squared in R#6. Outliers Revisited#---------------------------------------------------------------------------------# 1. Scatter Plots in R#--------------------------------------------------------------------------------- #Previously we've looked at various plots in R. #Today we are going to learn how to do a scatter plot in R.#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## Scatter Plot: plot(x=data$variable, y=data$variable)#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^##Let's start by reading in the lab8a.txt file. a=read.table('lab8a.txt', header=T)a #The data "a" contains variables named X and Y variables#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##Exercise 1-1: # A) Create a scatter plot for the variables in a. # Put X on the x-axis and Y on the y-axis # B) What does the scatter plot look like? Is it linear?#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #A) plot(y=a$Y, x=a$X)#B) #Looks kinda linear#---------------------------------------------------------------------------------# 2. Linear Regression in R#--------------------------------------------------------------------------------- #R has a function that computes the regression #of Y on X (Best fit line).#Linear Regression is just the function of y=Mx + b, #there is a slope and intercept.#In Linear Regression, we re-write this function as y=?x + a#??????????????????????????????????????????????????????????????##Thought Question 1: In the equation of y=?x + a, #which is the slope and which is the intercept term. #??????????????????????????????????????????????????????????????##^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## Linear Regression: lm(data$variable ~ data$variable)# lm(outcome/dependent variable ~ predictor/independent variable/determinant)#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^##If we wanted to find the best fit line for our data #we could use the linear regression function:lm(a$Y~a$X)#How can we interpret the values we get?#Intercept = -0.3436 #When x is zero, #the mean value of y is -0.3436#Slope = 1.1153 #For a 1 unit increase in x, y increases by 1.1153 points#??????????????????????????????????????????????????????????????##Thought Question 2: How would we interpret the slope if the #coefficent had been negative? eg. -1.1153#??????????????????????????????????????????????????????????????# #---------------------------------------------------------------------------------# 3. Outliers in Regression#--------------------------------------------------------------------------------- #One of the concerns we should have about the data in the # previous section is that there are outliers in the #original data. Let's trim the outliers to see#how this affects our regression lines.#I'll re-plot the dataplot(y=a$Y, x=a$X)#I'm also going to use a new command to identify#the rows where outliers occur.#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## Give Row info for plots: identify(data)#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^#identify(a)#On the plot, we can click on the outliers to figure out#what row the outliers occur on #Then they are row 20 and 8#Now, we can close the plot we created #and let's go back and plot our data, #but now by adding the regression line#To add the regression line, #I'll store the results of the linear regression #into an object called m1 (model1)m1=lm(a$Y~a$X)#We can then re-draw the plot plot(y=a$Y, x=a$X)#And use the abline() function to add the regression lineabline(m1)# Now, in order to see the effects of the outliers# I might like to see the regression lines from data #where the outliers have been removed. #I'll create some other versions of the dataset "a"#that does just that.a8=a[-8,] #Does not contain row 8a20=a[-20,] #Does not contain row 20a8_20=a[c(-8,-20),] #Does not contain row 8 and 20#I can then run the regressions on these limited datasets. m2=lm(a8$Y~a8$X)m3=lm(a20$Y~a20$X)m4=lm(a8_20$Y~a8_20$X)#And then plot all the regression lines on the plot.plot(y=a$Y, x=a$X)abline(m1, col="black")abline(m2, col="red")abline(m3, col="green")abline(m4, col="blue")#---------------------------------------------------------------------------------# 4. Hypothesis testing in Regression#---------------------------------------------------------------------------------#In regression, or goal in general is to find out#if two variables are related to each other #This is indicated to us when two variables #do not have a slope of 0.#Then, in regression, our Null and Alternative Hypotheses are:# H0: Beta_1 = 0# HA: Beta_1 different from 0#We can test the null hypothesis here by using #the "summary()" command on our MODELS#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## summary of results: summary(model)#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^##*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##Example: If I wanted to know if our original model #without removing outliers had slope of 0#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#summary(m1)#We then compare the p-value to our alpha level#A) If pval < alpha, then Reject the Null Hypothesis#B) If pval > alpha, then Fail to Reject the Null Hypothesis#I fail to reject the null hypothesis of Beta_1 = 0#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##Exercise 4-1: # Test the null hypothesis for the slopes in Models 2, 3, and 4. # Do you reject or fail to reject for each model? #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##A) If pval < alpha, then Reject the Null Hypothesis#B) If pval > alpha, then Fail to Reject the Null Hypothesissummary(m2) #Reject H0 p-value: 0.0141 compare to alpha=.05summary(m3) #Fail to reject H0summary(m4) #Reject H0#---------------------------------------------------------------------------------# 5. Correlation and R-squared in R#--------------------------------------------------------------------------------- #We just learned how to do Linear regression in R #using the lm() function.#Linear regression told us how a 1 unit increase in X#affects Y.#Correlation coefficents (rho) are another way of #representing how strong a linear relationship is.#They range from -1 to 1, with values further away #from zero representing a stronger association. #Positive values indicate that as X increases,#Y increases#Negative values indicate that as X increases, #Y decreases#Below is the function for a correlation between#two variables:#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## Correlation: cor(data$variable1, data$variable2)#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^##*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##Exercise 5-1: # Use the correlation function to find the correlation#between X and Y in our datset "a"#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#cor(a$X, a$Y)#Because the correlation is positive we know that # as X increases, Y increases. #We also knew this before when we did linear regression#and looked at the plots. #The correlation coefficent is related to something #from linear regression called R-squared.#R-squared represents the proportion of variability #in the outcome (Y) explained by the predictor (X). #IF we think of our correlation coefficent as R,#then R-squared will be:cor(a$X, a$Y)^2 #This means that ~14.6% of the variability in Y # is explained by the scores in X.#Which is the same value reported in the linear regressionsummary(m1)#---------------------------------------------------------------------------------# 6. Outliers Revisited#---------------------------------------------------------------------------------#For this part, we will need the Rallfun-v23.txt source file#Import the data from lab8b.txt into R in table form; save as object called b.b=read.table('lab8b.txt',header=TRUE)b #Contains 26 values, X variable and Y variable#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##Exercise 6-1: # A) Create a scatter plot (X on x-axis, Y on y-axis)# B) Based on the scatter should the correlation # be positive or negative?#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##A) #B) # Previously we visually identified outliers # and used the identify() command to find their # row numbers so we could eliminate them # Instead, let's use a more systematic approach#using an outlier removal technique called #the Mad-Median#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## Identify Outliers using Mad-Median: out(data$variable)#^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^## For example, I can identify the outliers in X by doing the following.out(b$X)# n.out tells me how many outliers their are# out.id tells me the rows they occur on.#I could then create a new version of b that does not contain outliers in XbrmX=b[c(-19,-25), ]#And then find the correaltion for this versioncor(brmX$Y, brmX$X)#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*##Exercise 6-2: # A) Create dataset brmY that is a new version of b but with outliers in Y removed (using Mad-Median)# B) Create dataset brmXY that is a new version of b but with outliers in X OR Y removed (using Mad-Median).# C) What is the correaltion coefficeint between X and Y for part A# D) What is the correaltion coefficeint between X and Y for part B#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #A)out(b$Y)brmY=b[c(-22,-26), ]#B)brmXY=b[c(-22,-26,-19,-25),]#C) cor(brmY$Y, brmY$X)#D)cor(brmXY$Y, brmXY$X)#Now, if we look at all these correlation values #removing these various outliers, what do we notice?cor(b$Y, b$X)cor(brmX$Y, brmX$X)cor(brmY$Y, brmY$X)cor(brmXY$Y, brmXY$X)#And now what does our plot look like if we removed outliers in X or Y plot(y=b$Y, x=b$X)points(y=brmXY$Y, x=brmXY$X,col="red")#Are there still outliers? #??????????????????????????????????????????????????????????????##Thought Question 3: What does this tell us about #our outlier detection technique?#??????????????????????????????????????????????????????????????#

1. The Area of a rectangle is represented by and its length is represented by .What is the width of the rectangle?If you double the length of the rectangle but keep the same width, what is the new area of the rectangle?2. Consider the following expression:Subtract the rational expressionsGiven the equation , solve the equation using your solution from part a.Solve the equation using LCD.Compare the methods used to solve the equation in parts c and d. Which method do you think is better and why?3. The average cost of electricity used for an 18 cubic ft. refrigerator that is 10 years old is $128 per year. A new refrigerator the same size costs about $81 per year for electricity. (Source: https://www.efficiencyvermont.com)Suppose a new refrigerator costs $500. Determine the total cost of having either refrigerator for 12 years, assuming the only costs associated with the refrigerator are its purchase cost and electricity.Determine the total annual cost for a new refrigerator that lasts 12 years, assuming the only costs associated with the refrigerator are its purchase cost and electricity.Write a function that gives the annual cost of a new refrigerator as a function of the number of years you own the refrigerator.What are the asymptotes of the rational function in part d? What does the horizontal asymptote mean in the context of this problem?Sketch the graph of the rational function from part c.If you plan on owning your refrigerator for 12 years, would it be more cost effective to keep your old refrigerator or purchase a new one? Explain.

An expert who works for a car magazine obtained random data (rounded to the nearest thousand) among two categories of used or new cars: Domestic ForeignThe expert would like to understand sales based on list price (rounded to the nearest thousand dollars), sale price (rounded to the nearest thousand dollars), and number of days it takes to sell each car. The complete data set is in the file named Cars.Managerial ReportPrepare a report (see below) that summarizes your assessment of the nature of the car market. Be sure to include the following seven items in your report.Calculate descriptive statistics (mean, median, range, standard deviation, and coefficient of variation) to summarize each of the three variables for the all Domestic cars.Use z-scores to determine if there any outliers in the data set for any of the three variables. If there are any outliers in any category, please list them and state for which category they are an outlier.If a result is an outlier, state whether it is below or above the mean.How can one use the above descriptive statistics to understand the Domestic car market better?Could there be any other descriptive statistics that can help one understand the Domestic car market better?Descriptive statistics (mean, median, range, standard deviation, and coefficient of variation) to summarize each of the three variables for the all Foreign cars.Use z-scores to determine if there any outliers in the data set for any of the three variables.If there are any outliers in any category, please list them and state for which category they are an outlier.If a result is an outlier, state whether it is below or above the mean.How can one use the above descriptive statistics to understand the Foreign car market better?Could there be any other descriptive statistic that can help one understand the Foreign car market better?Compare your summary results from #1 and #2. Discuss any specific statistical results that would help the car expert understand the two-car markets.Develop a 98% confidence interval estimate of the population mean sales price and population mean number of days to sell for Domestic cars. What is the margin of error?Interpret what each confidence intervals means. How can both confidence intervals provide useful car sales information?What other confidence interval would you choose to compute to obtain more information related to Domestic car sales? Explain why.What happens to the confidence interval as the level of confidence is increased?Develop a 98% confidence interval estimate of the population mean sales price and population mean number of days to sell for Foreign cars. What is the margin of error?Interpret what each confidence intervals means. How can both confidence intervals provide useful car sales information?What other confidence interval would you choose to compute to obtain more information related to Domestic car sales? Explain why.Assume the car expert requested estimates of the mean number of days to sell for the Domestic cars with a margin of error of seven days and the mean selling price of foreign cars with a margin of error of eight days.Using 98% confidence, how large should the sample sizes be for each one?How could the sample size formula be useful in understanding both types of car businesses? What are some advantages of a larger sample size?Suppose a Domestic car has a list price of $30,000 and a Foreign car has a list price of $30,000. What is your estimate of the final selling price (based on the percent difference for the sale and list price) and number of days required to sell each of these cars?How would these two estimates be useful in this application?Write a report that adheres to the formatting and APA expectations outlined on the CSU Global Writing Center (Links to an external site.). As with all written assignments at CSU Global, you should have in-text citations and a reference page.Submit your Excel file in addition to your report.Requirements:Paper must be written in third person.Your paper should be four to five pages in length (counting the title page and references page) and cite and integrate at least one credible outside source. The CSU Global Library (Links to an external site.) is a great place to find resources.Include a title page, introduction, body, conclusion, and a reference page.The introduction should describe or summarize the topic or problem. It might discuss the importance of the topic or how it affects you or society as a whole, or it might discuss or describe the unique terminology associated with the topic.The body of your paper should answer the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.The conclusion should summarize your thoughts about what you have determined from the data and your analysis, often with a broader personal or societal perspective in mind. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.Include any tables of data or calculations, calculated values, and/or graphs associated with this problem in the body of your assignment.Document formatting, citations, and style should conform to the CSU Global Writing Center (Links to an external site.). A short summary containing much that you need to know about paper formatting, citations, and references is contained in the New Sample APA Paper (Links to an external site.). In addition, information in the CSU Global Library (Links to an external site.)has many helpful areas (Writing Center, Writing Tips, Template Examples/Papers Essays, and others).