Concept explainers
Do taller adults make more money? The authors of the paper “Stature and Status: Height, Ability, and Labor Market Outcomes” (Journal of Political Economics [2008]: 499–532) investigated the association between height and earnings. They used the simple linear regression model to describe the relationship between x = Height (in inches) and y = log(Weekly gross earnings in dollars) in a very large sample of men. The logarithm of weekly gross earnings was used because this transformation resulted in a relationship that was approximately linear.
The paper reported that the slope of the estimated regression line was b = 0.023 and the standard deviation of b was sb = 0.004. Carry out a hypothesis test to decide if there is convincing evidence of a useful linear relationship between height and the logarithm of weekly earnings. Assume that the basic assumptions of the simple linear regression model are reasonably met.
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Chapter 13 Solutions
Introduction to Statistics and Data Analysis
- What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?arrow_forwardThis problem is inspired by a study of the “gender gap” in earnings in topcorporate jobs [Bertrand and Hallock (2001)]. The study compares totalcompensation among top executives in a large set of U.S. public corporations in the 1990s. (Each year these publicly traded corporations must report total compensation levels for their top five executives.)a. Let Female be an indicator variable that is equal to 1 for females and 0for males. A regression of the logarithm of earnings onto Female yields "ln (Earnings)" = 6.48 - 0.44 Female, SER = 2.65. (0.01) (0.05)i. The estimated coefficient on Female is -0.44. Explain what thisvalue means.ii. The SER is 2.65. Explain what this value means.iii. Does this regression suggest that female top executives earn lessthan top male executives? Explain.iv. Does this regression suggest that there is gender discrimination?Explain. b. Two new variables, the market value of the firm (a measure of firmsize, in millions of…arrow_forwardForensic scientists can learn about events at a crime scene by collecting data. Ex: Properties of glass shards at a crime scene such as chemical composition can indicate what type of glass was broken at the scene. Possible types include building glass (building windows or doors), vehicle glass (car windows or doors), or household glass (lightbulbs, baking dishes). The fitted logistic regression model for predicting whether a glass shard is building glass based on sodium is: = 20.02+(-1.42) (sodium) 1+e20.02+(-1.42) (sodium) Calculate the log-odds that a glass shard with sodium = 13.08 is building glass. Ex: 1.23 C Calculate the probability that a glass shard with sodium = 13.08 is building glass.arrow_forward
- The November 24, 2001, issue of The Economist published economic data for 15 industrialized nations. Included were the percent changes in gross domestic product (GDP), industrial production (IP), consumer prices (CP), and producer prices (PP) from Fall 2000 to Fall 2001, and the unemployment rate in Fall 2001 (UNEMP). An economist wants to construct a model to predict GDP from the other variables. A fit of the model GDP = , + P,IP + 0,UNEMP + f,CP + P,PP + € yields the following output: The regression equation is GDP = 1.19 + 0.17 IP + 0.18 UNEMP + 0.18 CP – 0.18 PP Predictor Coef SE Coef тР Constant 1.18957 0.42180 2.82 0.018 IP 0.17326 0.041962 4.13 0.002 UNEMP 0.17918 0.045895 3.90 0.003 CP 0.17591 0.11365 1.55 0.153 PP -0.18393 0.068808 -2.67 0.023 Predict the percent change in GDP for a country with IP = 0.5, UNEMP = 5.7, CP = 3.0, and PP = 4.1. a. b. If two countries differ in unemployment rate by 1%, by how much would you predict their percent changes in GDP to differ, other…arrow_forwardAnarrow_forwardThe rental of an apartment (?R) near campus is a function of the square footage (??Sq). A random sample of apartments near campus yielded the following summary statistics: ?¯R¯ = $340, ??¯Sq¯ = 93, ??sR = $ 29.1, and ???sSq= 10.5. Suppose also that the correlation between price and weight is ?r = 0.83. (a) Write the implied least squares linear regression equation. (b) Suppose an apartment has 75 sqft. Predict its price based on the above model. (c) Suppose the true rental of the apartment in part (b) is $ 345. What is the value of the residual?arrow_forward
- Zagat’s publishes restaurant ratings for various locations in the United States. The following table contains the Zagat rating for food, décor, service, and the cost per person for a sample of 100 restaurants located in New York City and in a suburb of New York City. Develop a regression model to predict the cost per person, based on a variable that represents the sum of the ratings for food, décor, and service. Predict the mean cost per person for a restaurant with a sum-mated rating of 50. What should you tell the owner of a group of restaurants in this geographical area about the relationship between the summated rating and the cost of a meal? Location Food Décor Service Summated Rating Coded Location Cost Bins Midpoints City 22 14 19 55 0 33 19.99 25 City 20 15 20 55 0 26 29.99 35 City 23 19 21 63 0 43 39.99 45 City 19 18 18 55 0 32 49.99 55 City 24 16 18 58 0 44 59.99 65 City 22 22 21 65 0 44 69.99 75 City 22 20 20 62 0 50 79.99 85 City 20 19…arrow_forwardSeven North American Green Frogs (Rana clamitans) had their jumping distance recorded (in mm) multiple times in a laboratory. The mean jumping distance for these frogs along with their length (measured from snout to vent in miMillimeters) are presented in the table below. Length of Frog 52 68 37 65 77 81 59 Mean Jumping Distance 546 673 415 659 793 814 563 (a) Determine the linear regression model that will best predit the mean jumping distance of a North American Green Frog based on the frog's length. (b) How well does the linear regression model fit this sample data? (c) Use the linear regression model to predict the mean jumping distance of a North American Green Frog that is 48 mm in length. No excel, please.arrow_forwardOnline clothes II For the online clothing retailer dis-cussed in the previous problem, the scatterplot of Total Yearly Purchases by Income showsThe correlation between Total Yearly Purchases and Incomeis 0.722. Summary statistics for the two variables are: a) What is the linear regression equation for predictingTotal Yearly Purchase from Income? b) Do the assumptions and conditions for regression ap-pear to be met? c) What is the predicted average Total Yearly Purchasefor someone with a yearly Income of $20,000? Forsomeone with an annual Income of $80,000?d) What percent of the variability in Total YearlyPurchases is accounted for by this model?e) Do you think the regression might be a useful one forthe company? Comment.arrow_forward
- The following scatterplot shows the selling price (according to Kelly’s Blue Book) of 31 Toyota Camry’s as a function of their mileage. a)What does this scatterplot tell you? The result of a simple linear regression analysis with “Selling Price” as the dependent and “Mileage” as the independent variable are given below: b) Is there a statistically significant relationship between selling price and mileage? Explain your answer. c) How would the selling price change for every additional 10,000 miles of a Camry? Find a 95% confidence interval for that change.arrow_forwardA study of emergency service facilities investigated the relationship between the number of facilities and the average distance traveled to provide the emergency service. The following table gives the data collected. Number ofFacilities AverageDistance(miles) 9 1.66 11 1.13 16 0.83 21 0.61 27 0.51 30 0.46 2. .Does a simple linear regression model appear to be appropriate? Explain. a.No, the scatter diagram suggests that there is no relationship. b.No, the scatter diagram suggests that there is a curvilinear relationship. c.Yes, the scatter diagram suggests that there is a linear relationship. 3.Develop an estimated regression equation for the data corresponding to a second-order model with one predictor variable. (Round your numerical values to four decimal places.)arrow_forwardWhat is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below: Least Squares Linear Regression of PRICE Interpret the standard deviation of the regression model. a) We expect most of the sampled diamond prices to fall within $1117.56 of their least squares predicted values. b) We can explain 89.25% of the variation in the sampled diamond prices around their mean using the size of the diamond in a linear model. c) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $1117.56. d) We expect most of the sampled diamond prices to fall within $2235.12 of their least squares predicted values.arrow_forward
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