Secondary Math Margin of Error

Transcription

1 Secondary Math Margin of Error

2 What you will learn: How to use data from a sample survey to estimate a population mean or proportion. How to develop a margin of error through the use of simulation models for random sampling.

3 Margin of Error A simulation (a trial) is only an approximation of the population parameter (the data used to describe the population). Each time a simulation is run, the results are slightly different. It is possible to give an interval that the population parameter falls within by finding a margin of error. A margin of error refers to the expected range of variation in a survey or simulation if it were to be conducted multiple times under the same procedures.

4 Margin of Error The margin of error is based on the sample size and the confidence level desired. The interval that includes the margin of error is called the confidence interval and is usually computed at 95% A confidence level of 95% means that you can be 95% certain that the actual population parameter falls within the confidence interval.

5 For example, the student body officers at your school conducted a survey to determine whether or not a majority of the students dislike the music played in the halls during class changes. They randomly interviewed students and determined that 52% of the students disliked the music with a margin of error of 3% that was calculated at a 95% confidence level. The confidence interval is 52% ±3% or 49% to 55%. It can be said that with 95% accuracy that between 49% and 55% of the students dislike music. Therefore, they really can t say that half of the school dislikes the music.

6 When calculating the means from several trials of a simulation the results are normally distributed. We can then use the fact that 95% of the data is within 2 standard deviations of the mean to find a margin of error with a 95% confidence level.

7 Vocabulary The margin of error accounts for the variation in results if the study or simulation were to be conducted multiple times under the same conditions. It does account for the random selection of individuals but does not account for bias. A confidence interval provides a range of plausible values for a population parameter. It can be found by using the sample statistic and the margin of error for the confidence level desired, i.e. sample statistic ± margin of error. The confidence level determines how likely it is that the actual population parameters falls on the confidence interval that was calculated using the sample statistic and the margin of error.

8 Calculating the Margin of Error with a 95% confidence level. If you are given a sample proportion, then the margin of error can be approximated by: 2 () where is the sample proportion and n is the sample size. If you are given sample mean and standard deviation, then the margin of error can be approximated by: 2 where s is the sample standard deviation and n is the sample size.

9 Calculating the Margin of Error with any confidence level. If you are given a sample proportion, then the margin of error can be approximated by: z () where is the sample proportion and n is the sample size. If you are given a sample mean and a standard deviation, then the margin of error can be approximated by: z where s is the sample standard deviation and n is the sample size. Remember that z (z-score) stands for how many standard deviations there are from the mean.

10 Calculating the Margin of Error with any confidence level. Let s say we want to determine the z-score to use for a confidence level of 90%. 90% would be the center of the normally distributed graph (area under the curve from 5% to 95%) If we find the z-score for the area below 5% we would know how many standard deviations 90% of the data would be away from the mean. Use your probability tables to determine what z- score would correlation with 5% or , thus you would multiply the formula by 1.64 instead of 2 for a 90% confidence level. Actual z-scores used for confidence levels: 80% = % = % = % = % = 2.576

11 What is the theoretical probability of getting any number on the spinner? = 1 5 = 0.2

12 a) Probability of the spinner landing on 2 Sample proportion: " 2 = # = 4 30 = Margin of error: 95% confidence interval: ± From to Theoretical probability: = would be within the confidence interval.

13 b) Probability of the spinner landing on 3 Sample proportion: " 2 = # = 6 30 = 0.2 Margin of error: 95% confidence interval: 0.2±0.146 From to Theoretical probability: = would be within the confidence interval.

14 c) Probability of the spinner landing on 5 Sample proportion: " 2 = # = 9 30 = 0.3 Margin of error: 95% confidence interval: 0.3±0.167 From to Theoretical probability: = would be within the confidence interval. Is it possible for it not to be in the confidence interval?

15 A fast food restaurant wanted to determine the wait times for customers in line. He timed the customers at random. a. Find the mean and standard deviation for the sample to the nearest tenth. Mean = 5.7, S = 2.2 b. Approximate the margin of error for a 95% confidence interval to the nearest tenth =0.8 30

16 A fast food restaurant wanted to determine the wait times for customers in line. He timed the customers at random. c. Find the 95% confidence interval. 5.7±0.8 From 4.9 minutes to 6.5 minutes d. Interpret the meaning of the interval in terms of wait times for customers. We can say with 95% confidence that the actual average wait time for a customer at the fast food restaurant is between 4.9 minutes and 6.5 minutes. e. What would the confidence interval be at a 99% level? = 1.035, 5.7 ± From to minutes.

17 What would happen to the interval if the confidence level was increased? Would it be wider, narrower, or shift? Wider you would have a larger margin of error. Think of the formula if you multiply by a larger number (higher the confidence level the further standard deviations away) then you will get a larger margin of error. What would happen to the interval if the sample size was increased? narrower more trials would be a smaller margin of error. Think about the formula if you are dividing by a larger number, your margin of error is going to smaller.