TEKS

2. TEKS • (6.10)  Probability and statistics. The student uses statistical representations to analyze data. The student is expected to: (B)  identify mean (using concrete objects and pictorial models), median, mode, and range of a set of data; • (7.11)  Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation. The student is expected to: (B)  make inferences and convincing arguments based on an analysis of given or collected data. • (7.15)  Underlying processes and mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions. The student is expected to: (B)  validate his/her conclusions using mathematical properties and relationships. • (8.13)  Probability and statistics. The student evaluates predictions and conclusions based on statistical data. The student is expected to: (B)  recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis.

3. NCTM Standards Grades 6-8 Expectations • Select and use appropriate statistical methods to analyze data - Find, use, and interpret measures of center and spread, including mean and interquartile range • Develop and evaluate inferences and predictions that are based on data - Use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken

4. Definitions to Know • Population - the entire group of individuals that we want information about • Sample - the part of the population that we actually examine in order to gather information • Random Sampling - a selection that is chosen randomly • Sampling Distribution - the probability distribution of a given statistic

5. More Definitions to Know • Confidence Interval - a statistical range with a specified probability that a given parameter lies within the range • Confidence Level - the level of certainty to which an estimate can be trusted • Standard Deviation - a measure of how spread out the data is • p-a known or given “true” proportion • p̂ - sample population proportion

6. Normal Distribution A basic fact of normal distribution is that 95% of all observations lie within two standard deviations on either side of the mean.

7. Normal Distribution So, if p̂ lies within two standard deviations of the true proportion in 95% of the samples, we can say that we are 95% confident that the unknown population proportion lies within a certain interval.

8. What do Confidence Statements Mean? “We got these numbers by a method that gives correct results 95% of the time.” The confidence interval can either – • Contain the true population proportion or • Not contain the true population proportion • We cannot know if our sample is one of the 95% for which the interval catches p or one of the unlucky 5%.

9. Significance Test • Used to assess whether an effect or difference is present in the population • Answers the question: “Is the observed effect larger than can reasonably be attributed to chance alone?” • Uses a correlation coefficient, r, to show if there is a relationship between the two variables and how strong it is

10. Steps for Calculating a 95% Confidence Interval • Calculate the mean, - Average data collected • Calculate the standard deviation, σ - Subtract the mean from every number to get the list of deviations - Square the resulting list of numbers - Add up all of the resulting squares to get their total sum - Find the mean of this sum, this is the variance - Find the square root of the variance

11. Steps for Calculating a 95% Confidence Interval • Calculate the standard deviation of the sampling distribution (standard error) = • Calculate the confidence interval = *1.96 comes from the Z-table and refers to the area of 2 standard deviations from the mean. 1.96 is always used for calculating the 95% confidence interval.

12. Demonstration Activity Use the data on the next slide to calculate an estimate for the true mean sales of the ten highest selling box office movies. Then create a confidence interval to back up your estimate.

13. Demonstration Activity Data retrieved from: The Internet Movie Database http://www.imdb.com/boxoffice/alltimegross

14. Calculating True Mean of Sales • Calculate the mean, • Calculate the standard deviation, σ - Subtract the mean from every number to get the list of deviations - Square the resulting list of numbers - Add up all of the resulting squares to get their total sum - Find the mean of this sum, this is the variance - Find the square root of the variance, yielding the standard deviation 448 millions of dollars 152, 82, 12, -8, -13, -18, -23, -48, -68, -68 23104, 6724, 144, 64, 169, 324, 529, 2304, 4624, 4624 42610 4261 65.28

15. Calculating Confidence Interval • Calculate the standard deviation of the sampling distribution = = • Calculate the confidence interval = 448 ± (1.96*20.64) = What is our confidence statement? 20.64 (407.55, 488.45) We are 95% confident that the true mean of sales for the ten highest selling box office movies is contained in the above confidence interval.

16. Possible Sources of Error • As noted at the bottom of the website: - Figures are not adjusted for inflation. - Some movies may still be in general release; all figures are estimated and subject to change.