Problem 3: Ten steel cables were tested in a laboratory by applying tensile forces of varying magnitudes. Deformations were observed as follows: Force (kg) 15 Increase in length (mm) 1.7 20 25 34 42 49 53 55 62 66 2.2 2.5 3.3 3.9 5.0 5.4 5.6 6.6 7.3 a. Estimate the parameters of a simple linear regression model with force as the independent variable. b. Find the 95% confidence interval for the two parameters. c. Test the hypothesis that the intercept is zero.
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- A researcher is investigating possible explanations for deaths in traffic accidents. He examined data from 2000 for each of the 52 cities randomly selected in the US. The data included information on the followingvariables: Deaths: The number of deaths in traffic accidents per city and Income: The median income per cityAs part of his study, he ran the following simple linear regression model as pictured : Question: Based on the above results, the researcher tested the hypotheses ( Null: B1=0 versus Alternative: B1 not equal to 0) using T test. What do we know about the test statistic of the test, what is the approximate p-value, and value of Rsquared? And based on your result, what is your conclusion? Show your work for full credit.An article presented data on compressive strength x and intrinsic permeability y of various concrete mixes and cures. Summary quantities are n = 14, Σy: = 572, Σy = 23530, Σ x₁ = 43, x² = 157.42, and xy; = 1697.80. Assume that the two variables are related according to the simple linear regression model. Round your answers to 3 decimal places. (a) Test for significance of regression using a = 0.05. (b) Estimate o2. i (c) In this model, what is the standard error of the intercept? i What is the standard error of the slope? iConsider the following population linear regression model of individual food expenditure: Y = 50 + 0.5X + u, where Y is weekly food expenditure in dollars, X is the individual’s age, and 50+0.5X is the population regression line. Suppose we generate artificial data for 3 individuals using this model. This artificial sample, which consists of 3 observations, is shown in the following table: Answer the following questions. Show your working. (a) What are the values of V1 and V4? (b) Suppose we know that in this artificial sample, the sample covariance between X and Y is 150, and the sample variance of X is 100. Compute the OLS regression line of the regression of Y on X. (Hint: Assume these summary statistics and the OLS regression line continue to hold in parts (c)-(e).) (c) What are the values of V5 and V7?
- A group of scientists and engineers aim to create fuel-efficient and fuel-efficient cars. In order to study the problem, they randomly selected a sample of 20 cars and took information from X: weight (hundreds of pounds) and Y: vehicle performance (mill / gal). Once the information was collected and analyzed, using a scatterplot, they determined that a linear model can fit the data. Using R the following information is obtained from the linear regression model. Y = 40.15−0.513X According to the model, what would be the weight of a car with a performance of 14 mill / gal? Select one: a. -39.98 lbs b. 39.98 lb c. 77.82 lb d. -77.82 lbThe table below shows the number of state-registered automatic weapons and the murder rate for several Northwestern states. xx 11.7 8.3 7 3.4 2.4 2.6 2.6 0.9 yy 14 11.1 10.1 6.7 6.1 6.4 6.5 4.9 xx = thousands of automatic weaponsyy = murders per 100,000 residents Determine the regression equation in y = ax + b form and write it below. (Round to 2 decimal places):____________________ A) How many murders per 100,000 residents can be expected in a state with 10.8 thousand automatic weapons? Round to 3 decimal places. Answer = B) How many murders per 100,000 residents can be expected in a state with 4.4 thousand automatic weapons? Round to 3 decimal places. Answer =Indicate the steps on creating letter b and the graphing tool used.
- The table below shows the number of state-registered automatic weapons and the murder rate for several Northwestern states. xx 11.7 8.1 6.7 3.3 2.7 2.3 2.2 0.3 yy 13.8 10.7 9.9 6.6 6.1 5.9 6.1 4.5 xx = thousands of automatic weaponsyy = murders per 100,000 residents Determine the regression equation in y = ax + b form and write it below. (Round to 2 decimal places) A) How many murders per 100,000 residents can be expected in a state with 6.4 thousand automatic weapons? Answer = Round to 3 decimal places. B) How many murders per 100,000 residents can be expected in a state with 6.7 thousand automatic weapons? Answer = Round to 3 decimal places.The peanut crop was harvested from five fields of various area. The following data are the mass of the crop from each field y (in kilograms) and the field area x (in hectares). y 7360 15760 13690 20080 12910 x 2.34 3.92 3.35 4.56 2.56 Round your intermediate answers to four decimal places (e.g. 98.7654).(a) Fit the simple linear regression model using the method of least squares. Find the estimate of σ2.Round your answer to the nearest integer (e.g. 9876).σ^2= (b) What change in the mean mass is expected when the field area changes by 1 hectare?Round your answer to the nearest integer (e.g. 9876).β^1= (c) Calculate the fitted value of y corresponding to x=3.92. Find the corresponding residual.Round your answer to the nearest integer (e.g. 9876).y^= Round your answer to the nearest integer (e.g. 9876).e= (d) Estimate the mean mass of the crop harvested from 4.5 hectares.Round your answer to the nearest integer (e.g. 9876).y^=Town A B C D E F GPopulation (lakh) (X ) 11 14 14 17 17 21 25No. of TV sets demanded ('000) (Y ) 15 27 27 30 34 38 46Fit a linear regression of Y on X and estimate the demand for TV sets for a city with apopulation of (a) 20 lakh and (b) 32 lakh.
- Q4: Effect of inereasing rate of shear of a pharmaceutical formulating on the resultant shearing stress, incorporating important parameters that are employed in linear regression analysis. Rate of shear (s) Theoretical shearing stress(Pa) Product of X and Y (X) (V) (XY) 20 40 40 4. 160 60 360 80 640 100 10 1000A study was conducted at Virginia Tech to determine if certain static arm-strengthmeasures have an influence on the “dynamic lift” characteristics of an individual.Twenty-five individuals were subjected to strength tests and then were asked toperform a weightlifting test in which weight was dynamically lifted overhead. The data are given here. a) Estimate ?0 and ?1 for the linear regression curve. b) Find a point estimate at 30. c) Plot the residuals versus the arm strength. Comment on the plot. You may use graphing tool. d) Compare (a) and (b)