Homework 7 Slide Set 8 Inference for Comparing Two Means Slide Set 9 Intro to Descriptive Linear Regression Details about submitting homework to Gradescope can be found in the homework assignment on Canvas 1 Highway Fuel Efficiency for Manual and Automatic Cars Each year the US Environmental Protection Agency EPA releases fuel economy data on cars manufactured in that year We took random samples of cars with manual and automatic transmissions manufactured in 2021 and recorded the highway fuel efficiency in miles gallon for each car We want to investigate the difference between the mean highway fuel efficiency mpg for cars with manual transmissions and the mean highway fuel efficiency mpg for cars with automatic transmissions Here are histograms of the data The summary statistics are Transmission Type Mean MPG Automatic Manual 23 68 30 88 SD 3 902 25 5 134 25 a What is the parameter of interest and what is the point estimate Be sure to include notation for both the parameter and the point estimate b Check the conditions for doing inference on the parameter of interest c Calculate a 99 confidence interval for the parameter and interpret that interval in context For your reference the critical value is 2 690 Homework 7 PSTAT 5LS Winter 2024 page 1 2 Body Temperatures Researchers at Stanford investigated several questions about the normal body temperature of adults in the United States Consider a random sample of healthy individuals from the Stanford study who were classified as either female or male In total we have data for 1000 patients 500 of whom were classified as female and 500 of whom were classified as male Below are histograms and summary statistics for body temperatures in the two groups of patients as male Let represent the difference between the population mean body temperatures for Is there a difference in the mean body temperatures for those classified as female and those classified those classified as female and the population mean body temperature for those classified as male a Describe two similarities and differences that you see in the distributions of body temperatures for b State the hypotheses to test whether there is a difference in the mean body temperatures for females group 1 and males group 2 You do not need to define the parameter c Assume that the conditions for inference are satisfied Calculate the test statistic Be sure to show d The p value for the test is 0 0047 Which of the following is a conclusion that can be made at the the two groups your work 0 05 level circle one i There is a difference between the mean body temperatures for those classified as female and ii There is not a difference between the mean body temperatures for those classified as female those classified as male and those classified as male iii Our analysis suggests that there is a difference between the mean body temperatures for those classified as female and those classified as male iv Our analysis does not suggest that there is a difference between the mean body temperatures for those classified as female and those classified as male Homework 7 PSTAT 5LS Winter 2024 page 2 3 Car Sales A random sample of 90 used cars for sale on an internet site was taken and the price in 1000s and mileage previous miles driven in 1000s were recorded Here are a scatterplot and regression output from R a What is the value of the correlation coefficient Be sure to show your work b Use what you see in the scatterplot below to describe the relationship between price and mileage Make sure to address all four aspects and to include context c Write the equation of the regression line Be sure to define your variables in the context of the question d Calculate the estimated price for a car that has 55 000 miles on it Note Remember that price is in 1000s and the mileage variable is in 1000s Be sure to include units for your final answer and to show your work e The car with 55 000 miles on it is priced at 26 700 Again remember that price is in 1000s and the mileage variable is in 1000s Calculate the residual for this car Be sure to include units for your final answer and to show your work f On their homework assignment one of your friends wrote that the estimate for the slope coefficient means that the price of a used car will decrease by 590 15 for each additional 1000 miles driven There is something not quite correct about what they said How should their statement be changed to make this a correct interpretation g Is it reasonable to use this linear model to estimate the price for a car that has 110 000 miles on it Clearly state yes or no and briefly in 1 2 sentences explain your reasoning Homework 7 PSTAT 5LS Winter 2024 page 3 4 Words and Sentences It makes sense that the more sentences there are in a newspaper article the more words there are Is there a positive linear relationship between the number of words and the number of sentences We randomly selected 346 articles from a local newspaper to find out Here is R output for the data a What is the value of the correlation coefficient for these data Show your work and clearly mark your final answer b Which of the following statements is a correct interpretation of the value 0 7566 in the output i On average there is a predicted increase of 0 7566 sentences for each increase of 1 word in an article in this newspaper ii On average there is a predicted increase of 0 7566 words for each increase of 1 sentence in iii iv an article in this newspaper About 75 7 of the variability in the number of words can be explained by its linear relationship with the number of sentences in an article in this newspaper About 75 7 of the variability in the number of sentences can be explained by its linear relationship with the number of words in an article in this newspaper c Write the equation of the regression line for this model Be sure to use proper notation and context d One of the articles in the sample has 55 sentences Use the regression line to estimate the number of words in the article Be sure to include units for your final answer and to show your work e The article that has 55 sentences actually has 1109 words Calculate the residual for this article Be sure to include units for your final answer and to show your work Homework 7 PSTAT 5LS Winter 2024 page 4