The value of the correlation coefficient can also be found

What two things should be done before one performs a regression analysis?

What are the assumptions for regression analysis?

What is the general form for the regression line used in statistics? y _ a _ bx

What is the symbol for the slope? For the y intercept? b, a

What is meant by the line of best fit?

When all the points fall on the regression line, what is the value of the correlation coefficient? r would equal _1 or _1.

What is the relationship between the sign of the correlation coefficient and the sign of the slope of the regression line?

As the value of the correlation coefficient increases from 0 to 1, or decreases from 0 to _1, how do the points of the scatter plot fit the regression line? They would be clustered closer to the line.

How is the value of the correlation coefficient related to the accuracy of the predicted value for a specific value of x? The closer r is to _1 or _1, the more accurate the predicted value will be.

If the value of r is not significant, what can be said about the regression line?

When the value of r is not significant, what value should be used to predict y? When r is not significant, the mean of the y values should be used to predict y.

Gas Tax and Fuel Use The gas tax and fuel use are shown. Tax 21.5 23 18 24.5 26.4 19 Usage 1062 631 920 686 736 684 Find y when x _ $0.25. Not significant so no regression should be done.

Commercial Movie Releases New movie releases per studio and gross receipts are as follows: No. of releases 361 270 306 22 35 10 8 12 21 Gross receipts (million $) 3844 1962 1371 1064 334 241 188 154 125 Find y when x_200 new releases. y _ 181.661_7.319x; y _ 1645.5 (million $)

Forest Fires and Acres Burned Number of fires and number of acres burned are as follows: Find \(y^\prime\) when x = 60.

Years and contribution data are as follows: Years x 1 5 3 10 7 6 Contribution y, $ 500 100 300 50 75 80 Find y when x _ 4 years. y _ 453.176 _ 50.439x; 251.42

State Debt and Per Capita Taxes Data for per capita state debt and per capita state tax are as follows: Per capita debt 1924 907 1445 1608 661 Per capita tax 1685 1838 1734 1842 1317 Find y when x _ $1500 in per capita debt. Not significant so no regression should be done.

School Districts and Secondary Schools The number of school districts and the number of secondary schools in the district are shown. School districts 53 19 24 17 95 68 Secondary schools 50 27 187 84 143 216 Find y when x _ 70. Since r is not significant, no regression should be done.

Triples and Home Runs The number of triples and the number of home runs obtained by a selected sample of MLB players are shown. Triples 25 23 51 19 20 43 Home runs 212 199 144 160 149 122 Find y when x _ 33. Since r is not significant, no regression should be done.

Egg Production Number of eggs and price per dozen are shown. Find y’ when x = 1600 million eggs. y’ = 1.252 -0.000398x; y’ = 0.615 per dozen

Emergency Calls and Temperature Temperature in degrees Fahrenheit and number of emergency calls are shown. Temperature x 68 74 82 88 93 99 101 No. of calls y 7 4 8 10 11 9 13 Find y when x _80_F. y __7.544 _0.190x; 7.656, or 8 calls

Faculty and Students The number of faculty and the number of students in a random selection of small colleges are shown. Faculty 99 110 113 116 138 174 220 Students 1353 1290 1091 1213 1384 1283 2075 Now find the equation of the regression line when x and y are interchanged. y _ _14.974 _ 0.111x

Precipitation and Snowfall/Sleet The number of days of precipitation and snowfall/sleet are shown. Precipitation 61 111 140 116 88 136 Snow/sleet 2 15 21 8 11 13 Find y when x _ 100 days. y _ _7.327 _ 0.175x; 10.173 in

Average Temperature and Precipitation Temperatures (in degrees Fahrenheit) and precipitation (in inches) are as follows: Avg. daily temp. x 86 81 83 89 80 74 64 Avg. mo. precip. y 3.4 1.8 3.5 3.6 3.7 1.5 0.2 Find y when x _ 70_F. y _ _8.994 _ 0.1448x; 1.1

NHLAssists and Total Points The number of assists and the total number of points for a sample of NHL scoring leaders are shown. Assists 26 29 32 34 36 37 40 Total points 48 68 66 69 76 67 84 Find y when x _ 30 assists. y _ 2.693 _ 1.962x; 62

Fat Calories and Fat Grams The number of fat calories and the number of saturated fat grams for a random selection of breakfast entrees are shown. Fat calories 190 220 270 360 460 540 Sat. fat (g) 9 8 13 17 23 27 Find y when x _ 400 fat calories. y _ _2.417 _ 0.055x; 19.6 grams

Tall Buildings Stories and heights of buildings data follow: Stories x 64 54 40 31 45 38 42 41 37 40 Heights y 841 725 635 616 615 582 535 520 511 485 Find y when x _ 44. y _ 206.399 _ 9.262x; 613.9

Hospital Beds Licensed beds and staffed beds data follow: Licensed beds x 144 32 175 185 208 100 169 Staffed beds y 112 32 162 141 103 80 118 Find y when x _ 44. y _ 22.659 _ 0.582x; 48.267

Fireworks and Injuries These data were obtained for the years 1993 through 1998 and indicate the number of fireworks (in millions) used and the related injuries. Predict the number of injuries if 100 million fireworks are used during a given year. Fireworks in use x 67.6 87.1 117 115 118 113 Related injuries y 12,100 12,600 12,500 10,900 7800 7000 Source: National Council of Fireworks Safety, American Pyrotechnic Assoc.

Farm Acreage Is there a relationship between the number of farms in a state and the acreage per farm? Arandom selection of states across the country, both eastern and western, produced the following results. Can a relationship between these two variables be concluded? No. of farms (thousands) x 77 52 20.8 49 28 58.2 Acreage per farm y 347 173 173 218 246 132 Source: World Almanac.

SAT Scores Educational researchers desired to find out if a relationship exists between the average SAT verbal score and the average SAT mathematical score. Several states were randomly selected, and their SAT average scores are recorded below. Is there sufficient evidence to conclude a relationship between the two scores? Verbal x 526 504 594 585 503 589 Math y 530 522 606 588 517 589 Source: World Almanac.

Coal Production These data were obtained from a sample of counties in southwestern Pennsylvania and indicate the number (in thousands) of tons of bituminous coal produced in each county and the number of employees working in coal production in each county. Predict the amount of coal produced for a county that has 500 employees. No. of employees x 110 731 1031 20 118 1162 103 752 Tons y 227 5410 5328 147 729 8095 635 6157

Television Viewers A television executive selects 10 television shows and compares the average number of viewers the show had last year with the average number of viewers this year. The data (in millions) are shown. Describe the relationship. Viewers last year x 26.6 17.85 20.3 16.8 20.8 Viewers this year y 28.9 19.2 26.4 13.7 20.2 Viewers last year x 16.7 19.1 18.9 16.0 15.8 Viewers this year y 18.8 25.0 21.0 16.8 15.3 Source: Nielsen Media Research.

Absences and Final Grades An educator wants to see how the number of absences for a student in her class affects the students final grade. The data obtained from a sample are shown. No. of absences x 10 12 2 0 8 5 Final grade y 70 65 96 94 75 82

Fathers and Sons Weights A physician wishes to know whether there is a relationship between a fathers weight (in pounds) and his newborn sons weight (in pounds). The data are given here. Fathers weight x 176 160 187 210 196 142 205 215 Sons weight y 6.6 8.2 9.2 7.1 8.8 9.3 7.4 8.6

Age and Net Worth Is a persons age related to his or her net worth? A sample of 10 billionaires is selected, and the persons age and net worth are compared. The data are given here. Age x 56 39 42 60 84 37 68 66 73 55 Net worth (billion $) y 18 14 12 14 11 10 10 7 7 5 Source: The Associated Press.

For Exercises 13, 15, and 21 in Section 101, find the mean of the x and y variables. Then substitute the mean of the x variable into the corresponding regression line equations found in Exercises 13, 15, and 21 in this section and find y . Compare the value of y with for each exercise. Generalize the results.

The y intercept value a can also be found by using the equation a _ y _ bx y Verify this result by using the data in Exercises 15 and 16 of Sections 101 and 102.

The value of the correlation coefficient can also be found by using the formula \(r=\frac{b s_x}{s_y}\) where \(s_x\) is the standard deviation of the x values and \(s_y\) is the standard deviation of the y values. Verify this result for Exercises 18 and 20 of Section 10-1.