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Describe the data as having a positive correlation, a negative correlation, or approximately no correlation. Tell whether the correlation coefficient for the data is closest to 21, 20.5, 0, 0.5, or 1.
a.
x
y
1
1
b.
x
y
1
1
Solution
a. The scatter plot shows a strong correlation. So, the best estimate given is r 5 .
b. The scatter plot shows a weak correlation. So, r is between and , but not too close to either one. The best estimate given is r 5 .
Example 1 Describe and estimate correlation coefficients
1.
x
y
1
1
2.
x
y
1
1
Checkpoint For the scatter plot, (a) tell whether the data has a positive correlation, a negative correlation,or approximately no correlation, and (b) tell whether the correlation coefficient for the data is closest to 21, 20.5, 0, 0.5, or 1.
The table below gives the number of people y who attended each of the first seven football games x of the season. Approximate the best-fitting line for the data.
x 1 2 3 4 5 6 7
y 722 763 772 826 815 857 897
1. Draw a .
4 65 7
6500
700
750
800
850
900
x
y
Nu
mb
er
of
pe
op
le
Football game
2 310
2. Sketch the best-fitting line.
3. Choose two points on the line. For the scatter plot shown, you might choose (1, ) and (2, ).
4. Write an equation of the line. The line that passes through the two points has a slope of:
m 5 5
Use the point-slope form to write the equation.
y 2 y1 5 m(x 2 x1) Point-slope form
y 2 5 Substitute for m, x1, and y1.
y 5 Simplify.
An approximation of the best-fitting line is y 5 .
Example 2 Approximate the best-fitting line
3. The table gives the average class score y on each chapter test for the first six chapters x of the textbook.
x 1 2 3 4 5 6
y 84 83 86 88 87 90
a. Approximate the best-fitting line for the data.
b. Use your equation from part (a) to4 65
820
84
86
88
90
x
y
Ave
rag
e c
lass s
co
re
Test
2 310
predict the average class score on the chapter 9 test.
10. Household Size The table shows the average household size y in the United States from 1930 to 2000. Draw a scatter plot of the data and describe the correlation shown. Let t represent the number of years since 1930.
Baseball The table shows the height of a baseball that is hit, with x representing the time (in seconds) and y representing the baseball’s height (in feet). Use a graphing calculator to find the best-fitting quadratic model for the data.
Time, x 0 2 4 6 8
Height, y 3 28 40 37 26
Enter the data into two lists Make a scatter plot of of a graphing calculator. the data.
L1 L2 L30 32 284 406 378 26
L1(1)=0
Use the quadratic regression Check how well the model fits the data by graphing the model and the data in the same viewing window.
model feature to find the best-fitting quadratic model for the data.
QuadReg
y 5 ax2 1 bx 1 c
a 5
b 5
c 5
The best-fitting quadratic model is
.
Example 4 Best-fitting quadratic model for data
4. Use a graphing calculator to find the best-fitting model for the data in the table.
In Exercises 10 and 11, use the following information.
Youth Football The table shows the number of participants y in a local youth football program from 2003 to 2008. Assume that t represents the number of years since 2003.
Year, t 0 1 2 3 4 5
Participants, y 24 28 33 41 54 74
10. Use a graphing calculator to fi nd the best-fi tting quadratic model for the data.
11. Using the model, how many participants are projected for 2011?
The lists show the number of memberships sold each month for one year by two competing athletic clubs. Compare the mean and standard deviation for the numbers of memberships sold by the two athletic clubs.
In Exercises 11 and 12, fi nd the mean, median, mode, range, and standard deviation of the data set.
11. Oil Change The data set below gives the waiting times (in minutes) for several people having the oil changed in their cars at an auto mechanics shop.
22, 18, 25, 21, 28, 26, 20, 28, 20
12. Hockey The data set below gives the numbers of goals for the 10 players who scored the most goals during the 2003–2004 National Hockey League regular season.
41, 41, 41, 38, 38, 36, 35, 35, 34, 33
13. Telephone Calls The data sets below give the lengths (in minutes) of long distance telephone calls made from a household during two months. Compare the mean and standard deviation for the calls made during the two months.
}x and standard deviation s. For a randomly selected x-value from the distribution, find P(}x 2 s ≤ x ≤ }x 1 2s).
SolutionThe probability that a randomly selected x-value lies between and is the shaded area under the normal curve. Therefore:
P(}x 2 s ≤ x ≤ }x 1 2s) 5 1 1
5
Example 1 Find a normal probability
1. A normal distribution has mean }x and standard deviation s. For a randomly selected x-value from the distribution, find P(x ≤ }x 2 s).
Checkpoint Complete the following exercise.
Math Scores The math scores of an
169 278 387 496 605 714 823
exam for the state of Georgia are normally distributed with a mean of 496 and a standard deviation of 109. About what percent of the test-takers received scores between 387 and 605?
Solution
The scores of 387 and 605 repressent standard deviation on either side of the mean. So, the percent of test-takers with scores between 387 and 605 is
7.4 PracticeA normal distribution has mean } x and standard deviation s. Find the indicated probability for a randomly selected x–value from the distribution.
1. P(x ≥ } x 2 s) 2. P(x ≤ } x 1 3s) 3. P(x ≤ } x 2 3s)
Give the percent of the area under the normal curve represented by the shaded region.
4.
s
x 2
3
s
x 2
2
s
x 2
x s
x 1
s
x 1
2
s
x 1
3
5.
s
x 2
3
s
x 2
2s
x 2
x s
x 1
s
x 1
2
s
x 1
3A normal distribution has a mean of 25 and a standard deviation of 5. Find the probability that a randomly selected x–value from the distribution is in the given interval.
6. Between 25 and 30 7. Between 15 and 25 8. Between 20 and 35
A normal distribution has a mean of 75 and a standard deviation of 10. Use the standard normal table of your textbook to fi nd the indicated probability for a randomly selected x–value from the distribution.
12. P(x ≤ 75) 13. P(x ≤ 85) 14. P(x ≤ 55)
15. P(x ≤ 87) 16. P(x ≤ 69) 17. P(x ≤ 45)
In Exercises 18 and 19, use the following information.
Breakfast A restaurant is busiest on Sunday from 6:00 A.M. to 9:00 A.M. During these hours, the waiting time for customers in groups of 5 or less to be seated is normally distributed with a mean of 20 minutes and a standard deviation of 4 minutes.
18. What is the probability that customers in groups of 5 or less will wait 8 minutes or less to be seated during the busy Sunday morning hours?
19. What is the probability that customers in groups of 5 or less will wait 24 minutes or more to be seated during the busy Sunday morning hours?
In Exercises 20 and 21, use the following information.
Light Bulbs A company produces light bulbs having a life expectancy that is normally distributed with a mean of 1800 hours and a standard deviation of 65 hours.
20. Find the z-score for a life expectancy of 2000 hours.
21. What is the probability that a randomly selected light bulb will last at most 2000 hours?
7.5 Select and Draw Conclusions from SamplesGoal p Study different sampling methods for
collecting data.
GeorgiaPerformanceStandard(s)
MM2D1a
Your NotesVOCABULARY
Population
Sample
Unbiased sample
Biased sample
Population mean
Margin of error
MARGIN OF ERROR FORMULA
When a random sample of size n is taken from a large population, the margin of error is approximated by:
Margin of error 5 6
This means that if the percent of the sample responding a certain way is p (expressed as a decimal), then the percent of the population that would respond the same
Assemblies A student wants to survey everyone at his school about the quality of the school's assemblies. Identify the type of sample described as a self-selectedsample, a systematic sample, a convenience sample, or a random sample.
a. The student surveys every 8th student that enters the assembly.
b. From a random name lottery, the student chooses 125 students and teachers to survey
Solutiona. The student uses a rule to select students, so the
sample is a sample.
b. The student chooses from a random name lottery, so the sample is a sample.
Example 1 Classify samples
1. A local mayor wants to survey local area registered voters. She mails surveys to the individuals that are members of her political party and uses only the surveys that are returned.
Checkpoint Identify the type of sample described.
Tell whether each sample in Example 1 is biased or unbiased. Explain your reasoning.
Solution
a. The sample is because the student surveys the students, but not the teachers.
b. The sample is because both students and teachers are surveyed.
Newspaper Survey In a survey of 325 students and teachers, 30% said they read the school's newspaper every weekday. (a) What is the margin of error for the survey? (b) Give an interval that is likely to contain the exact percent of all students and teachers who read the school's newspaper every weekday.
Solution
a. Margin of error 5 61
}Ï
}
n5 6
1ø
The margin of error for the survey is about %.
b. To find the interval, add and subtract %.
30% 2 % 5 %
30% 1 % 5 %
It is likely that the exact percent of all students and teachers who read the school's newspaper every weekday is between % and %.
Example 3 Find a margin of error
2. Tell whether the sample in Exercise 1 is biased or unbiased. Explain your reasoning.
3. In Example 3, suppose the sample size is 400 students and teachers. What is the margin of error for the survey?
Identify the type of sample described. Then tell if the sample is biased. Explain your reasoning.
1. A gym is conducting a survey to fi nd out how often members attend the gym each week. A gym employee asks every other person attending the gym on a particular weekend.
2. A clothing store wants to know the favorite seasons of its customers. Surveys are placed on a table for customers to fi ll out as they enter the store.
3. A company wants to know how often its employees use the company’s cafeteria for lunch. The company asks employees that have just fi nished eating lunch in the cafeteria on Friday.
Find the margin of error for a survey that has the given sample size. Round your answer to the nearest tenth of a percent.
7.6 Sample Data and PopulationsGeorgiaPerformanceStandard(s)
MM2D1a, MM2D1d
Your Notes
Goal p Collect sample data from populations.
A gym has 467 female members and 732 male members. The marketing director of the gym wants to form a random sample of 30 female members and a separate random sample of 60 male members to answer some survey questions. Each female member has a membership number from 1 to 467 and each male member has a membership number from 1001 to 1732. Use a graphing calculator to select the members who will participate in each random sample.
Random sample of female members:
Using the random integer feature of a graphing calculator to generate random integers between and produces the following sample answer.
The random sample of female members have membership numbers
.
Random sample of male members:
Using the random integer feature of a graphing calculator to generate random integers between and produces the following sample answer.
The random sample of male members have membership numbers
.
Example 1 Collect data by randomly sampling
1. In Example 1, suppose there are 245 female members and 532 male members. The marketing director wants to form a random sample of 12 female members and a separate random sample of 15 male members. Use a graphing calculator to select the members who will participate in each random sample.
2. In Example 2, suppose the population mean is 14.5 and the population standard deviation is 10.4. Compare the means and standard deviations of the random samples to the population parameters.
Checkpoint Complete the following exercise.
Homework
A company wants to know how many minutes it takes their employees to drive to work each day. Gillian and Ted, two employees, collect separate random samples. Their results are displayed below. The population mean is 18.4 and the population standard deviation is about 15.6. Compare the means and standard deviations of the random samples to the population parameters.
The mean of Gillian's sample is the population mean, while the mean of Ted's sample is the population mean. The standard deviations of both samples are the population standard deviation.
In Exercises 1–6, use a graphing calculator to generate fi ve random integers in the given range.
1. 1 to 100 2. 200 to 400 3. 101 to 450
4. 25 to 1000 5. 60 to 70 6. 1001 to 1765
For a large population, the mean is 11.2 and the standard deviation is about 8.4. Compare the mean and standard deviation of the random sample to the population parameters.
10. Movies Two students want to know the number of DVDs owned by each student in their school. Jake and Juan collect separate random samples. The population mean is 14.3 and the population standard deviation is about 6.7. Compare the means and standard deviations of the random samples to the population parameters.
7.7 Choose the Best Model for Two-Variable DataGoal p Choose the best model to represent a set of data.Georgia
PerformanceStandard(s)
MM2D2a, MM2D2c
Your Notes
Teachers' Salaries The table shows the teacher's salary y (in dollars) for a certain school district, where x is the number of years of teaching experience. Use a graphing calculator to find a model for the data.
x 1 2 3 4
y 30,624 32,436 34,167 35,989
x 5 6 7
y 37,684 39,311 41,098
1. Make a scatter plot. The points lie approximately on a .This suggests a model.
2. Use the regression feature to find an equation of the model.
3. Graph the model along with the data to verify that the model fits the data well.
A model for the data is y 5 .
Example 1 Use a linear model
1. Use a graphing calculator to find
x
y
10
1
a model for the data. Then graph the model and the data in the same coordinate plane.
Roller Coaster Riders A manager at a local amusement park kept a record of the number of people who ride the most popular roller coaster at the park. The table shows the number of people y who rode the roller coaster x hours after the park had opened. Use a graphing calculator to find a model for the data.
x 0 2 4 6 8 10 12
y 85 163 282 341 398 381 304
Solution1. Make a scatter plot. The points
form an .This suggests a model.
2. Use the regression feature to find an equation of the model.
3. Graph the model along with the data to verify that the model fits the data well.
A model for the data is y 5 .
Example 2 Use a quadratic model
2. Use a graphing calculator to find a model for the data. Then graph the model and the data in the same coordinate plane.
7.7 PracticeUse a graphing calculator to fi nd the equation that best models the data.
1. x 1 2 3 4 5 6 7
y 2 3 5 6 8 9 11
2. x 2 5 7 10 12 14 18
y 21 29 35 40 37 32 23
3. x 1 2 3 4 5 6 7
y 5 11 20 31 42 56 65
4. x 5 10 15 20 25 30 35
y 22 31 44 65 86 104 123
5. x 3 6 10 15 18 22 27
y 2 3 4 7 9 13 21
6. Drive-Thru Banking A bank records the length of time y (in minutes) that a customer has to wait each hour before getting service at the drive-thru window. The bank provides drive-thru service from 9:00 A.M. to 4:30 P.M. (x 5 1 represents 9:00 A.M.) Use the regression feature of a graphing calculator to fi nd a model for the data. If the bank extended its drive-thru hours, how long would a customer have to wait at 5:00 P.M.? Round your answer to the nearest whole minute.