Margin of Error. What is margin of error and why does it exist?

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Barnard Franklin
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1 Trig Honors Margin of Error Name: What is margin of error and why does it exist? Thanks to a network of GEO stationary and polar orbiting satellites providing better data, faster supercomputers, and improved modeling, forecasters at NOAA's National Hurricane Center have been able to reduce the margin of error in predicting a hurricane's path by almost half. In 2005 the average seasonal tracking error in a 48-hour forecast of where a storm would end up was 110 nautical miles. Now it's 65 miles. These graphics show the forecast path and estimated uncertainty for Hurricane Katrina with a three-day lead. On the left, a forecast run in 2015 uses more advanced techniques to narrow the predicted path of the storm. The original forecast from 2005 is shown on the right. The smaller size of the potential path (shown in the white area) in the 2015 version is a testament to the significant improvement of modeling and forecast capabilities since Katrina made landfall. 1. A college professor assigns reading for his students, but doesn t know how many pages they actually read each night. He conducted a simple random sample to determine how many pages his students read for each assignment. The professor found the sample mean to be 115 pages read with a margin of error of 2.5 pages. Express this result as an interval. Summary: 2. What questions could we ask about the professor s calculation? Commented [G1]: How many students were sampled? How were the students sampled? Would a different interval occur with a different sample? How did he compute the margin of error? Does the interval actually contain the true mean? How strong is his belief that this interval contains the true mean?

2 3. The diagram below shows levels of confidence that would determine the professor s interval. a. If the professor wanted to make his interval wider than the widest interval above, what % confidence could he use? b. If the professor wanted to make his interval narrower than the narrowest interval above, what % confidence could he use? Is this something a statistician would do? ** For the purposes of our class, we will always give you the margin of error at a % confidence level. If you are interested in learning more, take AP Stats in the future! ** 4. Students at Hinsdale Central were randomly sampled to determine how many hours of sleep students get each night. The sample mean was 6.2 and the margin of error at the 95% confidence level was 1.4. a. Express the 95% confidence interval of the mean. b. Is the true population mean guaranteed to be within your interval above? 5. Women were randomly sampled at a local park to determine how many hours a week they spend running. The sample mean was 3.7 and the margin of error at the 90% confidence level was 0.6. Express the 90% confidence interval of the mean. 6. Fifty-five percent of students at a high school like the color green. Given the margin of error at the 80% confidence level is 0.27, express the 80% confidence interval of the proportion.

3 7. [17.15, 23.84] expresses a 95% confidence interval of the mean number of inches of rainfall in Boston in March. a. Calculate the sample mean. b. Calculate the margin of error. 8. [0.14, 0.76] expresses the 68% confidence interval of the proportion of students who play two sports. a. Calculate the sample proportion. b. Calculate the margin of error. What does it mean to have a 90% confidence interval? Commented [G2]: Discuss here how samples of the same size can offer wide and narrow intervals. Some samples can have data values very spread out and others can be very close together, but both are trying to represent the population parameter. Just because the interval is narrow does not mean that it describes the parameter. An example would be average running time for the mile for students at HC. If you sample the track team, it would probably be a narrow interval, but far from the average. If you sample athletes, it would be better, but better still would be to randomly sample students in the school. What it is not:

4 9. Below are two simulated sampling distributions for the sample proportion of female students in random samples from all the students at Union High School. a. Based on the two sampling distributions above, what do you think is the population proportion of female students? b. Discuss the spread of each sampling distribution. c. One of the sampling distributions above is based on random samples of size 30, and the other is based on random samples of size 60. Which sampling distribution corresponds to the sample size of 30? Explain your choice. d. How is margin of error effected by the sample size? 10. The college professor (from example 1) wants to make his confidence interval smaller. a. What can he do to make the interval smaller? b. Which method maintains the confidence of the interval containing the true mean?

5 Trig H Margin of Error Homework 1. Based on random samples of river heights, oceanographers estimate that 4,800 kilometers is discharged into the Arctic Ocean annually. The estimate has a margin of error of 240 km at the 90% confidence level. Which of the following is the most reasonable claim about the volume of freshwater discharged annually into the Arctic Ocean? a. Between 4800 and 5040 kilometers b. Between 4560 and 5040 kilometers c. Between 240 and 4800 kilometers d. Between 240 and 4320 kilometers 2. A random sample of international flights in 2013 showed that 79% percent of the flights arrived within 15 minutes of their scheduled arrival time, and this estimate had a margin of error of 3% percent at a 95% percent confidence level. Which of the following conclusions is best? a. Exactly 79% arrived within 15 minutes of their scheduled times. b. Exactly 76% arrived within 15 minutes of their scheduled times. c. Between 76% and 82% arrived within 15 minutes of their scheduled times. d. Between 92% and 98% arrived within 15 minutes of their scheduled times. 3. In a poll of 578 randomly selected American adults, 44.8% of the respondents said that airlines should allow in-flight calls on airplanes. The poll reported a margin of error of 2.5% at a 95% percent confidence level. Which of the following is most likely to be equal to the percentage of all American adults who would say that airlines should allow in-flight calls? a. 40% b. 43% c. 48% d. 95% 4. A research assistant randomly selected 75 undergraduate students from the list of all students enrolled in the psychology-degree program at a large university. She asked each of the 75 students, How many minutes per day do you typically spend reading? The mean reading time in the sample was 89 minutes, and the margin of error for this estimate was 4.28 minutes. Another research assistant intends to replicate the survey and will attempt to get a smaller margin of error. Which of the following samples will most likely result in a smaller margin of error for the estimated mean time students in a psychologydegree program read per day? a. 40 randomly selected undergraduate psychology-degree program students b. 40 randomly selected undergraduate students from all degree programs at the college c. 300 randomly selected undergraduate psychology-degree program students d. 300 randomly selected undergraduate students from all degree programs at the college 5. [0.8, 10.2] expresses a 90% confidence interval of the mean number of days spent at the beach during the summer. a. Calculate the sample mean. b. Calculate the margin of error.

6 6. A random sample of 35 four-door passenger vehicles had a mean gas mileage in miles per gallon of The estimate had a margin of error of 2.6 mpg at a 98% confidence level. Of the following, which is the most plausible value for the true mean gas mileage of all four-door passenger vehicles? a. 24 b. 29 c. 32 d The following image represents 20 samples taken of the same size. Describe what this picture means in the context of statistics. 8. Cities in the United States were sampled to determine the average number of minutes people drive to work. The 90% confidence interval for the mean was determined to be [5.6, 53.7]. a. What could be done to make the margin of error smaller? b. What could be done to make the confidence interval wider? In a peer-reviewed study of 1000 Labrador dogs, 80% could jump 1 meter in the air in 19 out of 20 attempts. There was a 2.5% margin of error when measuring the height of jumps. 9. Julia wants to know her Labrador s chances of jumping 1 meter in the air. Based on the peer-reviewed study, which one of the following statements could be correct? a. Julia s Labrador has a 95% chance of jumping 1 meter in the air. b. Julia s Labrador has an 80% chance of jumping 1 meter in the air. c. Julia s Labrador has an 80% chance of jumping 1 meter in the air 95% of the time it jumps. d. Julia s Labrador has an 80% chance of jumping at least meters in the air 95% of the time it jumps. 10. Hans owns a bulldog whose weight and height are almost exactly half of the weight and height of Julia s Labrador. Given this information, which of the following statements is true? a. Han s bulldog would have about a 40% chance of jumping at least meters in the air. b. Han s bulldog would have an 80% chance of jumping 0.5 meters in the air. c. Han s bulldog would have a 95% chance of jumping 0.5 meters in the air. d. The peer-reviewed study cannot be used to predict the jumping height of Hans s bulldog.

What Is a Sampling Distribution? DISTINGUISH between a parameter and a statistic

Section 8.1A What Is a Sampling Distribution? Learning Objectives After this section, you should be able to DISTINGUISH between a parameter and a statistic DEFINE sampling distribution DISTINGUISH between