# [Table] A real-estate agent is trying to determine the relationship between the distance a 3-bedroom home is from New York City and its average selling price. He records the data for 6 homes shown below. Linear or Quadratic? Equation: Approximate cost of home 90 miles from NYC?

Question
Linear equations and graphs
[Table]
A real-estate agent is trying to determine the relationship between the distance a 3-bedroom home is from New York City and its average selling price. He records the data for 6 homes shown below.
Equation:
Approximate cost of home 90 miles from NYC?

2021-02-10
Input the x-values under L1 in a graphing calculator and the yy-values, in thousands, under L2 (for example, input 755,000 as 755). Then graph the scatterplot to determine if it's linear or quadratic:
[Graph]
The points approximately lie on a straight line so the data is linear.
Use the LinReg feature on your graphing calculator to find the values of aa and bb for the line of best fit:
[Graph]
Since a≈−4.3 and b≈795.6, the equation is y=−4.3x+795.6.
If a home is x=90 miles from NYC, then y=−4.3(90)+795.6=408.6≈409 (round to the nearest whole number since the costs in the table are rounded to the nearest thousand). The cost of the home is then about $409,000 ### Relevant Questions asked 2020-12-25 Case: Dr. Jung’s Diamonds Selection With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring. After a bit of searching, Dr. Jung located a number of earring sets that he would consider purchasing. But he knew the pricing of diamonds varied considerably. To assist in his decision making, Dr. Jung decided to use regression analysis to develop a model to predict the retail price of different sets of round-cut earrings based on their color, clarity, and carat scores. He assembled the data in the file Diamonds.xls for this purpose. Use this data to answer the following questions for Dr. Jung. 1) Prepare scatter plots showing the relationship between the earring prices (Y) and each of the potential independent variables. What sort of relationship does each plot suggest? 2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why? 3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics? 4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase? 5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why? 1 6) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics? 7) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must actually square the model’s estimates to convert them to price estimates.) Which sets of earring appears to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase? 8) Dr. Jung now also remembers that it sometimes helps to include interaction terms in a regression model—where you create a new independent variable as the product of two of the original variables. Modify your spreadsheet to include three new independent variables, X4, X5, and X6, representing interaction terms where: X4 = X1 × X2, X5 = X1 × X3, and X6 = X2 × X3. There are now six potential independent variables. If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why? 9) Suppose Dr. Jung decides to use color (X1), carats (X3) and the interaction terms X4 (color * clarity) and X5 (color * carats) as independent variables in a regression model to predict the square root of the earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics? 10) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. (Remember, your model estimates the square root of the earring prices. So you must square the model’s estimates to convert them to actual price estimates.) Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase? asked 2021-03-11 An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by $$\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.$$ $$\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$ ​(A) Complete the table below. $$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$ ​(Round to one decimal place as​ needed.) $$A. 20602060xf(x)$$ A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate. $$B. 20602060xf(x)$$ Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate. $$C. 20602060xf(x)$$ A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate. $$D.20602060xf(x)$$ A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate. ​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29 $$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ and for 35 $$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ The mileage for the tire pressure $$\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ is The mileage for the tire pressure $$\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is (Round to two decimal places as​ needed.) (D) Write a brief description of the relationship between tire pressure and mileage. A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase. B. As tire pressure​ increases, mileage decreases. C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease. D. As tire pressure​ increases, mileage increases. asked 2020-12-27 The scatter plot below shows the average cost of a designer jacket in a sample of years between 2000 and 2015. The least squares regression line modeling this data is given by $$\widehat{y}=-4815+3.765x.$$ A scatterplot has a horizontal axis labeled Year from 2005 to 2015 in increments of 5 and a vertical axis labeled Price ($) from 2660 to 2780 in increments of 20. The following points are plotted: $$(2003, 2736), (2004, 2715), (2007, 2675), (2009, 2719), (2013, 270)$$. All coordinates are approximate. Interpret the y-intercept of the least squares regression line. Is it feasible? Select the correct answer below: The y-intercept is −4815, which is not feasible because a product cannot have a negative cost. The y-intercept is 3.765, which is not feasible because an expensive product such as a designer jacket cannot have such a low cost. The y-intercept is −4815, which is feasible because it is the value from the regression equation. The y-intercept is 3.765 which is feasible because a product must have a positive cost.
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
Researchers have asked whether there is a relationship between nutrition and cancer, and many studies have shown that there is. In fact, one of the conclusions of a study by B. Reddy et al., “Nutrition and Its Relationship to Cancer” (Advances in Cancer Research, Vol. 32, pp. 237-345), was that “...none of the risk factors for cancer is probably more significant than diet and nutrition.” One dietary factor that has been studied for its relationship with prostate cancer is fat consumption. On the WeissStats CD, you will find data on per capita fat consumption (in grams per day) and prostate cancer death rate (per 100,000 males) for nations of the world. The data were obtained from a graph-adapted from information in the article mentioned-in J. Robbins’s classic book Diet for a New America (Walpole, NH: Stillpoint, 1987, p. 271). For part (d), predict the prostate cancer death rate for a nation with a per capita fat consumption of 92 grams per day. a) Construct and interpret a scatterplot for the data. b) Decide whether finding a regression line for the data is reasonable. If so, then also do parts (c)-(f). c) Determine and interpret the regression equation. d) Make the indicated predictions. e) Compute and interpret the correlation coefficient. f) Identify potential outliers and influential observations.
Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of $$\alpha = 0.05$$. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.) Lemons and Car Crashes Listed below are annual data for various years. The data are weights (metric tons) of lemons imported from Mexico and U.S. car crash fatality rates per 100,000 population [based on data from “The Trouble with QSAR (or How I Learned to Stop Worrying and Embrace Fallacy),” by Stephen Johnson, Journal of Chemical Information and Modeling, Vol. 48, No. 1]. Is there sufficient evidence to conclude that there is a linear correlation between weights of lemon imports from Mexico and U.S. car fatality rates? Do the results suggest that imported lemons cause car fatalities? $$\begin{matrix} \text{Lemon Imports} & 230 & 265 & 358 & 480 & 530\\ \text{Crashe Fatality Rate} & 15.9 & 15.7 & 15.4 & 15.3 & 14.9\\ \end{matrix}$$
Predicting Land Value Both figures concern the assessed value of land (with homes on the land), and both use the same data set. (a). Which do you think has a stronger relationship with value of the land-the number of acres of land or the number of rooms n the homes? Why? b. Il you were trying to predict the value of a parcel of land in e arca (on which there is a home), would you be able to te a better prediction by knowing the acreage or the num- ber of rooms in the house? Explain. (Source: Minitab File, Student 12. "Assess.")
True or False
1.The goal of descriptive statistics is to simplify, summarize, and organize data.
2.A summary value, usually numerical, that describes a sample is called a parameter.
3.A researcher records the average age for a group of 25 preschool children selected to participate in a research study. The average age is an example of a statistic.
4.The median is the most commonly used measure of central tendency.
5.The mode is the best way to measure central tendency for data from a nominal scale of measurement.
6.A distribution of scores and a mean of 55 and a standard deviation of 4. The variance for this distribution is 16.
7.In a distribution with a mean of M = 36 and a standard deviation of SD = 8, a score of 40 would be considered an extreme value.
8.In a distribution with a mean of M = 76 and a standard deviation of SD = 7, a score of 91 would be considered an extreme value.
9.A negative correlation means that as the X values decrease, the Y values also tend to decrease.
10.The goal of a hypothesis test is to demonstrate that the patterns observed in the sample data represent real patterns in the population and are not simply due to chance or sampling error.
The mill Mountain Coffee shop blends coffee on the premises for its customers. it sells three basic blends in 1- pound bags, Special , Mountain dark, and Mill regular. It uses four different types of coffee to produce the blends- Brazilian, mocha,Columbian, and mild. The shop used the following blend recipe requirements :
$$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}\text{Blend}&\text{Mix requirement}&\text{Selling price/lb(\\)}\backslash{h}{l}\in{e}\text{special}&\text{at least 40% columbian,}&{6.50}\backslash&\text{at least 30% mocha}\backslash{h}{l}\in{e}\text{Dartk}&\text{at least 60% Brazillian}&{5.25}\backslash&\text{no more than 10% mid}\backslash{h}{l}\in{e}\text{Regular}&\text{no more than 60% mid}&{3.75}\backslash&\text{at least 30% Brazillian}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$
The cost of Brazilian coffee is 2.00 per pound, the cost of mocha is $2.75 per pound, the cost of Columbian is$2.90 per pound,and the cost of mild is \$1.70 per pound. The shop has 110 pounds of Brazilan coffee. 70 pounds of mocha, 80 pounds of Columbian, and 150 pounds of mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize profit.
a. Formulate a linear programming model
b. Solve this model
Two scatterplots are shown below.
Scatterplot 1
A scatterplot has 14 points.
The horizontal axis is labeled "x" and has values from 30 to 110.
The vertical axis is labeled "y" and has values from 30 to 110.
The points are plotted from approximately (55, 60) up and right to approximately (95, 85).
The points are somewhat scattered.
Scatterplot 2
A scatterplot has 10 points.
The horizontal axis is labeled "x" and has values from 30 to 110.
The vertical axis is labeled "y" and has values from 30 to 110.
The points are plotted from approximately (55, 55) steeply up and right to approximately (70, 90), and then steeply down and right to approximately (85, 60).
The points are somewhat scattered.
Explain why it makes sense to use the least-squares line to summarize the relationship between x and y for one of these data sets but not the other.
Scatterplot 1 seems to show a relationship between x and y, while Scatterplot 2 shows a relationship between the two variables. So it makes sense to use the least squares line to summarize the relationship between x and y for the data set in , but not for the data set in .
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