94 Concept Byte Margin of Error

Margin of Error

The mean of a sample may or may not be the mean of the population the sample was drawn from. The margin of error helps you find the interval in which the mean of the population is likely to be. The margin of error is based on the size of the sample and the confidence level desired.

A 95% confidence level means that the probability is 95% that the true population mean is within a range of values called a confidence interval. It also means that when you select many different large samples from the same population, 95% of the confidence intervals will actually contain the population mean.

The means of all the samples follow a normal distribution.

0�1�1.96 1.96113.5% 13.5%

34% 34%

The normal distribution above shows that 95% of values are between -1.96 and 1.96 standard deviations from the mean. To find the margin of error based on the mean of a large set of data at a 95% confidence level, you use the formula ME = 1.96 # s

1n where ME is the margin of error, s is the standard deviation of the sample data, and n is the number of values in the sample. The confidence interval for the population mean m (pronounced myoo) is x - ME … m … x + ME, where x is the sample mean.

A grocery store manager wanted to determine the wait times for customers in the express lines. He timed customers chosen at random.

1. What is the mean and stardard deviation of the sample? Round to the nearest tenth of a minute.

2. At a 95% confidence level, what is the approximate margin of error? Round to the nearest tenth of a minute.

3. What is the confidence interval for a 95% confidence level?

4. What is the meaning of the interval in terms of wait times for customers?

11

Waiting Time (minutes)

3.3

7.5

3.4

4.7

5.4

3.1

5.1

4.6

3.5

4.6

5.9

3.2

5.2

6.2

8.2

4.7

6.7

8.2

6.7

5.5

4.2

4.5

6.5

2.5

7.3

3.6

3.8

9.7

8.2

4.8

CC-21 MACC.912.S-IC.2.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

MP 1

Common Core State Standards

HSM15_A2Hon_SE_CC_21_TrKit.indd 94 8/5/13 6:54 PM

Concept Byte Margin of Error 95

You can also find the margin of error and the confidence interval for a sample proportion. A sample proporton p

]

is the ratio xn, where x is the number of times an

event occurs in a sample of size n.

What is the sample proportion for each situation? Write the ratios as percents rounded to the nearest tenth of a percent.

5. In a poll of 1085 voters selected randomly, 564 favor Candidate A.

6. A coin is tossed 40 times, and it comes up heads 25 times.

To find the margin of error for a sample proportion at a 95% confidence level, use the

formula ME = 1.96 # Ap(1 - p)

n , where ME is the margin of error, p

]

is the sample

proportion, and n is the sample size. The confidence interval for the population proportion p is p - ME … p … p + ME.

Find (a) the sample proportion, (b) the margin of error, and (c) the 95% confidence interval for the population proportion.

7. In a survey of 530 randomly selected high school students, 280 preferred watching football to watching basketball.

8. In a simple random sample of 500 people, 342 reported using social networking sites on the Internet.

ExercisesFor Exercises 9–10, find the 95% confidence interval for the population mean or population proportion, and interpret the confidence interval in context.

9. A consumer research group tested the battery life of 36 randomly chosen batteries to establish the likely battery life for the population of same type of battery.

10. In a poll of 720 likely voters, 358 indicate they plan to vote for Candidate A.

11. Data Collection Roll a number cube 30 times. Record the results from each roll. In parts (a) and (b), find the sample proportion, the margin of error for a 95% confidence level, and the 95% confidence interval for the population proportion.

a. rolling a 2

b. rolling a 3

c. Is the 95% confidence intervals for the population proportion about the same for rolling a 2 and for rolling a 3?

d. Compare your sample proportions to the theoretical proportions for parts (a) and (b). Would you expect the theoretical proportion to be within the confidence intervals you found? Explain.

22

33

] ]

] ]

Battery Life(In Hours)

63.2

62.1

79.4

66.3

61.2

61.4

68.3

86.2

71.4

84.6

81.8

75.2

74.5

74.5

83.6

82.2

64.3

63.6

78.4

63.6

54.1

71.6

72.4

75.6

59.3

72.7

59.6

85.8

64.2

73.4

60.1

81.3

74.1

47.6

71.8

68.1

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94 Common Core

AnswersActivity 1 1. 5.4; 1.8 2. 0.6 3. 5.4 { 0.6 4. At a 95% confidence level, you

can say that the waiting time for customers at the grocery store is between 4.8 and 6.0 minutes.

Guided Instruction PURPOSE To find the margin of error and limits for 95% confidence intervals.PROCESS Students will•usetheformulaME = 1.96 # s

1n to find the

margin of error for 95% confidence intervals.•calculatethesampleproportionforasampling

situation.•findthemarginoferrorandconfidenceintervalforapopulationproportion.

DISCUSS Tell students what it means for an estimate to be within a margin of error. Ask, “If you make an estimate of an unknown quantity, would you want the margin of error to be large or small?” Explainthatconfidencelevelreferstohowsureyouare that your estimate is within the margin of error, and that larger margins of error lead to greater confidence levels.

Activity 1In this activity, students find a 95% confidence intervalforthemeanfromasampleofcustomerwait times.

Mathematical Practices CC-21supportsstudentsinmakingsenseofproblems,Mathematical Practice 1.

Q Whydothewaittimesinthetablerepresentasampleandnotapopulation? [They do not include all wait times.]

Q Whatistheprobabilitythattheactualpopulationmean will lie outside of the confidence interval? [5%]

94 Concept Byte Margin of Error

Margin of Error

The mean of a sample may or may not be the mean of the population the sample was drawn from. The margin of error helps you find the interval in which the mean of the population is likely to be. The margin of error is based on the size of the sample and the confidence level desired.

A 95% confidence level means that the probability is 95% that the true population mean is within a range of values called a confidence interval. It also means that when you select many different large samples from the same population, 95% of the confidence intervals will actually contain the population mean.

The means of all the samples follow a normal distribution.

0�1�1.96 1.96113.5% 13.5%

34% 34%

The normal distribution above shows that 95% of values are between -1.96 and 1.96 standard deviations from the mean. To find the margin of error based on the mean of a large set of data at a 95% confidence level, you use the formula ME = 1.96 # s

1n where ME is the margin of error, s is the standard deviation of the sample data, and n is the number of values in the sample. The confidence interval for the population mean m (pronounced myoo) is x - ME … m … x + ME, where x is the sample mean.

A grocery store manager wanted to determine the wait times for customers in the express lines. He timed customers chosen at random.

1. What is the mean and stardard deviation of the sample? Round to the nearest tenth of a minute.

2. At a 95% confidence level, what is the approximate margin of error? Round to the nearest tenth of a minute.

3. What is the confidence interval for a 95% confidence level?

4. What is the meaning of the interval in terms of wait times for customers?

11

Waiting Time (minutes)

3.3

7.5

3.4

4.7

5.4

3.1

5.1

4.6

3.5

4.6

5.9

3.2

5.2

6.2

8.2

4.7

6.7

8.2

6.7

5.5

4.2

4.5

6.5

2.5

7.3

3.6

3.8

9.7

8.2

4.8

CC-21 MACC.912.S-IC.2.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

MP 1

Common Core State Standards

HSM15_A2Hon_SE_CC_21_TrKit.indd 94 8/5/13 6:54 PM

Concept Byte Margin of Error 95

You can also find the margin of error and the confidence interval for a sample proportion. A sample proporton p ] is the ratio x

n, where x is the number of times an event occurs in a sample of size n.

What is the sample proportion for each situation? Write the ratios as percents rounded to the nearest tenth of a percent.

5. In a poll of 1085 voters selected randomly, 564 favor Candidate A.

6. A coin is tossed 40 times, and it comes up heads 25 times.

To find the margin of error for a sample proportion at a 95% confidence level, use the

formula ME = 1.96 # Ap(1 - p)

n , where ME is the margin of error, p ] is the sample

proportion, and n is the sample size. The confidence interval for the population proportion p is p - ME … p … p + ME.

Find (a) the sample proportion, (b) the margin of error, and (c) the 95% confidence interval for the population proportion.

7. In a survey of 530 randomly selected high school students, 280 preferred watching football to watching basketball.

8. In a simple random sample of 500 people, 342 reported using social networking sites on the Internet.

ExercisesFor Exercises 9–10, find the 95% confidence interval for the population mean or population proportion, and interpret the confidence interval in context.

9. A consumer research group tested the battery life of 36 randomly chosen batteries to establish the likely battery life for the population of same type of battery.

10. In a poll of 720 likely voters, 358 indicate they plan to vote for Candidate A.

11. Data Collection Roll a number cube 30 times. Record the results from each roll. In parts (a) and (b), find the sample proportion, the margin of error for a 95% confidence level, and the 95% confidence interval for the population proportion.

a. rolling a 2

b. rolling a 3

c. Is the 95% confidence intervals for the population proportion about the same for rolling a 2 and for rolling a 3?

d. Compare your sample proportions to the theoretical proportions for parts (a) and (b). Would you expect the theoretical proportion to be within the confidence intervals you found? Explain.

22

33

] ]

] ]

Battery Life(In Hours)

63.2

62.1

79.4

66.3

61.2

61.4

68.3

86.2

71.4

84.6

81.8

75.2

74.5

74.5

83.6

82.2

64.3

63.6

78.4

63.6

54.1

71.6

72.4

75.6

59.3

72.7

59.6

85.8

64.2

73.4

60.1

81.3

74.1

47.6

71.8

68.1

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CC-21 95

Activity 2 5. 52.0% 6. 62.5%

Activity 3 7. a. 52.8% b. 4.3% c. 48.5% … P … to 57.1% 8. a. 68.4% b. 4.1% c. 64.3% … P … 72.5%

Exercises 9. Thesamplemeanis70.5hours.The

95% confidence interval for the populationmeanis67.5 … m … 73.5, which tells you that, with 95% probability,themeanlifeforthebatterypopulationiswithinthestatedinterval.

10. Thesampleproportionis49.7%.The 95% confidence interval forthepopulationproportionis46.0% … P … 53.4%, which tells youthat,with95%probability,thepopulationproportionforvotersin the election is within the stated interval.

11a–c. Check students’ work. d. Answersmayvary.Sample:

Because the confidence interval is95%,youwouldexpectthetheoreticalprobability,theratio16, to be in the interval.

Activity 2Inthisactivity,studentscalculatethesampleproportionfordifferentsamplingsituations.

Activity 3Inthisactivity,studentsuseasampleproportionto find the margin of error and 95% confidence intervalforthepopulationproportion.

ERROR PREVENTION

Be sure students understand that they need to be consistentintheiruseofpercentsanddecimalswhenfindingconfidenceintervalsforsampleproportions.Iftheyconvertthesampleproportionfromadecimaltoapercent,thenthemarginoferrormustalsobeconvertedtoapercentbeforesubtracting and adding to find the limits of the confidence interval.

Q Whatisanothertermforsampleproportion? [relative frequency]

Q What does the confidence interval tell you about theactualpopulationproportion? [That there is a 95% chance that the population proportion is in the confidence interval.]

Q How do you think you would adjust the margin of error to write a 99% confidence interval for thepopulationproportion? [Increase the margin of error. (The z-score factor is 2.58 for a 99% confidence interval instead of 1.96.)]