Margin of Error = Confidence interval:

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1 NAME: DATE: Algebra 2: Lesson 16-7 Margin of Error Learning 1. How do we calculate and interpret margin of error? 2. What is a confidence interval 3. What is the relationship between sample size and margin of error? Inferential statistics are used to draw conclusions about a population using a sample. Once we have created our sampling distribution and arrived at our best estimate of the parameter of the population, we need to reveal just how good this estimate may be. Instead of stating the estimate as a single value, we form intervals surrounding the estimate. The margin of error is a measurement of how accurate we believe our sample statistic to be relative to the population. We will always be using a 95% margin of error. In any normal distribution, how much of the data falls within two standard deviations of the mean? Since roughly of all normally distributed data fall within two standard deviations of the mean, we use two standard deviations to develop the margin of error. Margin of Error = A Confidence Interval is a range of values we are fairly sure our true value lies in. We then use the margin of error to create a confidence interval Confidence interval: Example: A recent poll found that 36% of all respondents would vote for candidate A in an election. The poll reported a margin of error of 4%. Give an interpretation of what this margin of error means in terms of the 36% support for Candidate A.

2 Sample Size vs. Margin of Error If we increase the sample size, what happens to the standard deviation? If the standard deviation decreases, what happens to the margin of error? Therefore, a larger sample size will create a margin of error. Determining the Margin of Error Margin of Error = 2 standard deviation 1) At the beginning of the school year, school districts implemented a new physical fitness program. A student project involves monitoring how long it takes tenth graders to run a mile. A random sample of 20 students was taken. The sample mean time it currently takes tenth graders to run a mile was 7.9 minutes. a. The students doing the project collected random samples of students each and calculated the sample means. The standard deviation of their distribution of sample means was minutes. Based on this standard deviation, what is the margin of error for their sample mean estimate? b. Interpret the margin of error you found in part (b) in the context of this problem.

3 2) A candidate for political office commissioned a poll. His staff received responses from 900 likely voters and 55% of them said they would vote for the candidate. The staff then conducted a simulation of 1000 more polls of 900 voters, assuming that 55% of voters would vote for their candidate. The output of the simulation is shown in the diagram below. In a histogram, since the standard deviation is not given, estimate where 95% of the data seems to lie. Given this output, and assuming a 95% confidence level, the margin of error for the poll is closest to (1) 0.01 (2) 0.03 (3) 0.06 (4) 0.12 Determining the Confidence Interval Confidence interval: mean ± margin of error 3) An ecologist wants to know what percent of the 10,000 fish in a lake are cod. She takes 500 samples of size 50. The average for all these samplings is 0.24 with a standard deviation of This is the histogram of the sampling distribution of the sample proportion. Using this data, with a 95% confidence interval, we can determine that the percent of fish in the lake that are cod is which of the following? (1) Between 0.12 and 0.36 (3) Between 0.20 and 0.28 (2) Between 0.14 and 0.34 (4) Exactly 0.24

4 4) A concert promoter wants to estimate the average age of the 20,000 people attending a Taylor Swift concert. He takes 150 samplings of 80 people. The average of the means of all the samplings is 25.5 and the standard deviation is 1.5. The following is a histogram of the sampling distribution of the sample mean. Based on this data, with a 95% confidence interval, the researchers can determine that the average of the entire 20,000 person population is which of the following? (1) Exactly 25.5 (2) Between 22.5 and 28.5 (3) Between 23.5 and 27.5 (4) Between 24.5 and 26.5 Making Conclusions Using Confidence Intervals Determine the margin of error Use this to create a confidence interval See if the outcome given is within the confidence interval o If it is within the interval, it is a usual occurrence ( likely or fair ) o If it is not within the interval, it is an unusual occurrence. ( unlikely or unfair ) 5) Anne has a coin. She does not know if it is a fair coin. She flipped the coin 100 times and obtained 73 heads and 27 tails. She ran a computer simulation of 200 samples of 100 fair coin flips. The output of the proportion of heads is shown below. Given the results of her coin flips and of her computer simulation, which statement is most accurate? 1) 73 of the computer's next 100 coin flips will be heads. 2) 50 of her next 100 coin flips will be heads. 3) Her coin is not fair. 4) Her coin is fair.

5 6) An Algebra 2 class conducts a survey of a random sample of 50 students to determine what percent of the student body lives in a household where the annual income is over $60,000. According to their survey, 42% of the students live in such a household. The students conduct a series of simulations to determine the margin of error for this sample proportion. The results of the simulations lead the students to conclude that the actual percent of students who live in families with an income of over $60,000 is 42% ± 8%. Based on this margin of error, it is most unlikely that this percent of students live in households earn over $60,000 per year. (1) 33% (2) 48% (3) 42% (4) 50% 7) In a random sample of 250 men in the United States, age 21 or older, 139 are married. The graph below simulated samples of 250 men, 200 times, assuming that 139 of the men are married. a) Based on the simulation, create an interval in which the middle 95% of the number of married men may fall. Round your answer to the nearest integer. b) A study claims "50 percent of men 21 and older in the United States are married." Do your results from part a contradict this claim? Explain.

6 8) Fifty-five students attending the prom were randomly selected to participate in a survey about the music choice at the prom. Sixty percent responded that a DJ would be preferred over a band. Members of the prom committee thought that the vote would have 50% for the DJ and 50% for the band. A simulation was run 200 times, each of sample size 55, based on the premise that 60% of the students would prefer a DJ. The approximate normal simulation results are shown below. Using the results of the simulation, determine a plausible interval containing the middle 95% of the data. Round all values to the nearest hundredth. Members of the prom committee are concerned that a vote of all students attending the prom may produce a split. Explain what statistical evidence supports this concern.