Part 1: Short-term Forecasting: (i) Use a simple line plot of both time series to detect seasonal, irregular, or trend behaviors if any. Write a summary of your observations of both time series in your report. (ii) Perform exponential smoothing to forecast both prices for period 253. Use successive values of 0.15, 0,36, 0.57, and 0.78 for the smoothing parameter α. Next, calculate the MAPD (Mean Absolute Percentage Deviation) of each forecast; and based on the MAPDs, determine the value of α that has yielded the most accurate forecast for each stock. In your report, describe your results; and explain why in your opinion such values of α have yielded the most accurate forecasts for the two stocks. (iii) Use your exponential smoothing forecast of part (ii) with 𝜶=𝟎.56 and perform an adjusted exponential smoothing to forecast both prices for period 253. Use successive values of 0.15, 0.26, 0.47, and 0.88 for the trend parameters β for both stocks. Next, calculate the MAPEs (Mean Absolute Percentage Error) of your forecasts and determine the values of β that have provided the most accurate forecasts for both stocks. In your report, describe your results and explain why, in your opinion, such values of β have yielded the most accurate forecasts. part 2 (i) For each stock, use a 5-period weighted moving averages to forecast its value during periods 1 through 100. Use the weights 0.4 (for the most recent period), 0.2 (for the period before the most recent), 0.2 (for two periods ago), 0.1(for three periods ago) and 0.1(for four periods ago). Next, use the observed value for period 101 as the base of a linear trend, and use that linear trend to forecast the values of both stocks for periods 101 through 257. Write a summary of your results in your report. Describe how accurate this method of forecasting has been by comparing the forecasted values for periods 253-257 with their actual “Close” values on those specific days (Hint: check the actual values on https://finance.yahoo.com). (ii) Calculate the MAPEs (Mean Absolute Percentage Error) of your forecasts in question (i) above and compare them with the values obtained for your forecasts in Part 1. For each stock, describe which method has yielded a most accurate forecast. part 3For each stock, use simple regression of stock values versus the time periods to predict its values for periods 1 through 257. In your report, describe how the accuracy of this prediction has been compared to the methods used in Parts 1 and 2 of this project. (ii) Perform a residual analysis of your simple regression to verify whether regression is appropriate to use for each of the given data. In particular, determine: ● Whether the residuals are independent ● Whether the residuals are homoscedastic. ● Whether the residuals are normally distributed by plotting a Normal probability plot of the residuals ● Whether the residuals are normally distributed by performing a Chi-squared test for Normality of the residuals. After completing parts 1-3 and in your report, respond to the following question.

#Name: #Student ID:  rm(list=ls()) source('Rallfun-v33.txt') #PART 1 #A company claims that, when exposed to their toothpaste, 45% of all bacteria related to gingivitis are killed, on average. You run 10 tests and ???nd that the percentages of bacteria killed in each test was: #  38%, 44%, 62%, 72%, 43%, 40%, 43%, 42%, 39%, 41% #  Assuming normality, you will test the hypothesis that the average percentage of bacteria killed was 45% at alpha=0.05. #1.1) Write out the Null and Alternative hypotheses #1.2) Calculate the T-statistic and use Method 1 (we saw in class) to determine if the average bacteria killed was 45%. Do it by "hand". #Hint: Method 1 is to compare T to a critical value "c". #1.3) Do you reject or fail to reject the null?  ################################################ #PART 2 #Now, let's not assume normality #2.1) Using the same data as in Part 1, test the hypothesis that the 20% trimmed mean is 45%? #2.2) Do you reject or fail to reject the null? #2.3) Assuming your test in 2.1 is the truth, what type of error did you make in #1.3? ################################################ #PART 3 #In a study of court administration, the following times to disposition (in minutes) were determined for twenty cases and found to be: # 42, 90, 84, 87, 116, 95, 86, 99, 93, 92, 121, 71, 66, 98, 79, 102, 60, 112, 105, 98 #Assuming normality, you will test the hypothesis that the average time to disposition was 99 minutes at alpha=0.05. #3.1) Write out the Null and Alternative hypotheses #3.2) Calculate the T-statistic and use Method 2 (we saw in class) to determine if the average time to disposition was 99? Do it by "hand". #Hint: Method 2 is to evaluate the confidence interval. #3.3) Do you reject or fail to reject the null?  ################################################ #PART 4 #Now, let's not assume normality #4.1) Using the same data as in Part 3, test the hypothesis that the 20% trimmed mean is 99? #4.2) Do you reject or fail to reject the null? #4.3) Assuming your test in 4.1 is the truth, what type of error did you make in #3.3? ################################################ #PART 5 #Suppose you run an experiment, and observe the following values: #  12, 20, 34, 45, 34, 36, 37, 50, 11, 32, 29 #You will test the hypothesis that the average was 25 at alpha=0.05. #5.1) Write out the Null and Alternative hypotheses. Conduct the hypothesis test assuming normality. Use the "t.test" function. Do you reject or fail to reject the null? #5.2) Conduct the hypothesis test without assuming normality. Do you reject or fail to reject the null? #5.3) Assuming the answer in #5.2 is the truth, what type of error (if any) did you make in #5.1 by assuming normality? ------------------------------------------------------------------------------------------ Lab 7- Lecture Notes (FOR YOUR REFERENCE) #Lab 7-Contents #1. Formulating Hypotheses #2. T-statistics by Hand #3. Alpha Level #4. Evaluating Our Results  #5. Using the t.test function #6. T-tests with Trimmed Means (trimci function) #7. Type 1 and Type 2 Errors # Last week we talked about computations for when the Population  #Variance is known and unknown. # Given that we rarely know the population variance, #we will use the T-distribution for all of today's lab. #We will primarily work with the dataset brfss09_lab7.txt: ######################################################################################################################### #Behavioral Risk Factors Surveilance Survey 2009 (BRFSS09) Data Dictionary: #------------------------------------------------------------------------------------------------------------------------ #id: "Subject ID"Values[1,998] #physhlth: "# Days past month phsycial health poor" Values[1,30]  #menthlth: "# Days past month mental health poor"Values[1,30]  #hlthplan: "Have healthcare coverage?"Values 1=Yes, 2=No #age:"Age in Years"Values[18,99] #sex:"Biologic Sex"Values 0=Female, 1=Male #fruit_day: "# of servings of fruit per day"Values[0,20] #alcgrp: "Alcohol Consumption Groups"Values 1=None, 2= 1-2 drinks/day 3= 3 or more drinks/day #smoke:"Smoking Status"Values 0=Never, 1=Current EveryDay, 2=Current SomeDays, 3=Former #bmi:"Body Mass Index"Values[14,70] #mi:"Myocardial Infarction (heart attack)"Values 0=No, 1=Yes #------------------------------------------------------------------------------------------------------------------------ # For today's lab, let's start by reading in our datafile # 'brfss09_lab7.txt' into an object called mydata mydata=read.table('brfss09_lab7.txt', header=T) #This file contains: dim(mydata)#100 Subjects, 11 variables #With the following variables: names(mydata) # We have collected this data and would like to know #if the values we have found in our sample are different #from the reported values in the literature. # For example, it has been reported that the average BMI # in the population is 27.5. We would like to know if the  #values in our sample are somehow different than this value.  #--------------------------------------------------------------------------------- # 1. Formulating Hypotheses #--------------------------------------------------------------------------------- #Step 1 of determining if our BMI values differ from the  #national average of 27.5 is to formulate our hypotheses #We have TWO hypotheses #1) The Null Hypothesis: H0: mu = 27.5 #2) The Alternative Hypothesis: HA: mu != 27.5 #NOTE: mu=Population Mean  #The above hypotheses are Two-Sided.  #By this I mean that we are looking to see if our sample values of #BMI are greater than (>) OR less than (<) 27.5.  # A one-sided hypothesis test would look like: #H0: mu < 27.5 #HA: mu > 27.5 #OR #H0: mu > 27.5 #HA: mu < 27.5 #We will always use two-sided tests in this class,  #and similarly in the real world two-sided tests dominate. #Once we have our hypotheses we will evaluate them  #and determine one of two outcomes: # A) Reject the Null Hypothesis # B) Fail to Reject the Null Hypothesis #--------------------------------------------------------------------------------- # 2. T-statistics by Hand (well..with help from the computer) #--------------------------------------------------------------------------------- #Recall from the last lab, that the formula for a T-statistic is: # T = (SampleMean - PopMean) / (SampleSD/sqrt(N)) #Another way to write this would be: # T = (xbar - mu) / (s/sqrt(N)) #In this instance PopMean (mu) is the NULL hypothesis  #value we are testing against. #We can solve for the other values that we don't yet know: mu=27.5 xbar=mean(mydata$bmi) #28.22 s=sd(mydata$bmi) #6.32 N=100 T = (xbar - mu) / (s/sqrt(N)) T #1.14 #We end up with a T value of ~ 1.14 #But how does this tell us if our mean is different from 27.5 ???!!! #Before we move on, I want us to think about why we need  #to evaluate if our mean of 28.22 is different from 27.5. #Certainly we can see that these are different numbers,  #so what are we really asking here? #One way to think about it is that we are asking if our  #sample mean of 28.22 is different from 27.5 simply due to chance. #Think of a coin tossing example: #Your friend tosses a coin in the air and it lands on heads #3 times in a row! #While, kinda cool, seems like that is probably random chance. #What about if it landed on heads 100 times in a row?! #You would probably think she was cheating somehow! #Though it is possible to have 100 heads in a row  #by chance alone, it is very unlikely  #The point at which we say that something is random vs not  #is determined by our alpha level. #--------------------------------------------------------------------------------- # 3. Alpha Level #--------------------------------------------------------------------------------- # The alpha level is determined a priori (a head of time) #and used to set the threshold by which we consider something #to be random chance # A common alpha level is 0.05.  # We typically reject the null (think something is not chance)  #when the result we have (eg. 28.22) would only be #that extreme < 5% of the time by chance.  #Recall from Lab 6, that we use the alpha level  #to help figure out critical values (c) # c=qt(1-(alpha/2), df) #--------------------------------------------------------------------------------- # 4. Evaluating our Results #--------------------------------------------------------------------------------- # There are 3 ways to evaluate if our mean of 28.22  # is different from the null of 27.5 # All three ways will yield the same conclusion. #1) Compare T to a critical value (c) #2) Evaluate the Confidence interval  #3) Compare the p value to our alpha level ########################################################### #1)  Compare T to a critical value (c) #In order to compute the critical value (c),  #we must know the alpha level.  #We will choose a value of 0.05 (which is standard) alpha=0.05 df=100-1 c=qt(1-(alpha/2), df) #We can then compare the abosulte value of T (|T|)  #to the critical value c #A) If |T| > c, then Reject the Null Hypothesis #B) If |T| < c, then Fail to Reject the Null Hypothesis #Let's look at T can c  abs(T) c #What decision do we make about the Null Hypothesis???? ########################################################### #2)  Evaluate the Confidence interval #Rather than compare T to c,  #we could instead compute the confidence interval. #Recall the formula for the Confidence interval is: #LB= xbar - c*(s/sqrt(N)) #UB= xbar + c*(s/sqrt(N)) LB = xbar - c*(s/sqrt(N)) UB= xbar + c*(s/sqrt(N)) #A) If mu is not within the Confidence Interval,  #then Reject the Null Hypothesis #B) If mu is within the Confidence Interval,  #then Fail to Reject the Null Hypothesis #Let's look at LB and UB LB UB mu #What decision do we make about the Null Hypothesis???? ########################################################### #3)  Compare the p value to our alpha level #Lastly, we could find the probability value (or p-value)  #for the T statistic we created. #We can do this by using the pt() function we learned #about last week in lab 6. #There is a forumla for computing P values from T-statitics: # pval = 2*(1-pt(abs(T), df)) pval = 2*(1-pt(abs(T), df)) #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 #Let's look at our p-value. pval alpha #What decision do we make about the Null Hypothesis???? #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #Exercise 4-1:  #Evaluate if the mean age from our sample (mydata) is different #than the populatiuin mean age of 56 # A) Write down the Null and Alternative Hypotheses # B) Calculate the T-statistic by hand # C) Evaluate the Null hypothesis by using ALL 3 methods that  #    we just discussed # D) Based on the results in C, do you Reject or Fail to Reject  #    the Null Hypothesis? #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #A) #B)  #C) #Method 1: Compare T to a critical value (c) #Method 2: Evaluate the Confidence interval #Method 3: Compare the p value to our alpha level #D) #--------------------------------------------------------------------------------- # 5. Using the t.test function #--------------------------------------------------------------------------------- #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^# # One Sample T-Test : t.test(data$variable, mu) #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^# #It was really awesome that we figured out T by hand!  #And then figured out the confidence intervals and P values! #From now on, let's just use a program to do all this for us. #The function t.test will presume an alpha level of 0.05 by default. t.test(mydata$age, mu=56) # t.test(mydata$bmi, mu=27.5) #Much simpler! #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #Exercise 5-1: Use the t.test function to evaluate if  #A) the mean days of physical health (physhlth) is different  #    than the population mean of 10? Reject the Null? #B) the mean fruits per day (fruit_day) is different than  #   the populatiuin mean of 4?  Reject the Null? #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #A) #B) #--------------------------------------------------------------------------------- # 6. T-test with Trimmed Means #--------------------------------------------------------------------------------- #To use the T-test with trimmed means, #we will need to load in the source code 'Rallfun-v33.txt' #The trimmed mean T-test is beneficial in that it does not #presume a perfect Normal Distribution #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^# # Trimmed Mean T-Test: # trimci(data$variable, tr=0.2, alpha=0.05, null.value=0) #^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^# #For example, if I wanted to test if the age was equal to 56  #using Trimmed Means I could do: trimci(mydata$age, null.value=56) #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #Exercise 6-1: Use the trimci function to evaluate if  #A) the 20% trimmed mean of days of physical health (physhlth) is #  different than the populatiuin mean of 10? Reject the Null? #B) the 20% trimmed mean fruits per day (fruit_day) is different #than the populatiuin mean of 4?  Reject the Null? #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #A) #B) #--------------------------------------------------------------------------------- # 7. Type 1 and Type 2 Errors #--------------------------------------------------------------------------------- #Notice that we had very different answers to the same  #questions in Ex. 5-1 and 6-1 #Depending upon the method that we used. #This brings us to discussing Type 1 and Type 2 Error #A Type 1 error is when our test tells us to reject the null, #but in truth we should not have #A Type 2 error is when our test tells us to fail to reject the #null, but in truth we should have rejected the null #The following 2x2 square might make this easier to see. #           Truth #------------------------------------ #|    H0      |     HA  | #-------------- |-------|-----------| #My Test:    H0      | H0 Type 2| #-------------- |-------|-----------| #My Test: HA Type 1  | HA      | #------------------------------------ #For the next exercise, let's presume that our test of the trimmed mean is the Truth #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #Exercise 7-1:   #A) What type of error did we make when evaluating the mean  #of physhlth in exercise 5-1?  #B) What type of error did we make when evaluating the mean #of fruit_day in exercise 5-1? #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# #A)  #B)