40S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Statistics Lesson: ST-6 Confidence Intervals Confidence Intervals Learning Outcome B-4 ST-L6 Objectives: To perform Confidence Interval and Margin of Error Calculations.

40S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Statistics Lesson: ST-6 Confidence Intervals When we use statistical data to make a prediction (eg. the number of people in the class that prefer Pepsi is 12) we know that our prediction might not be exactly correct. Instead of predicting an exact value, we can predict a range (eg. the number of people in the class that prefer Pepsi is between 10 and 14). A confidence interval is a statement of this range of values (low number, high number). A 95 percent confidence interval states low and high values (that we can calculate) that are 95 percent certain capture the actual prediction. A 95 percent confidence interval is the same as data that are considered "... accurate 19 times out of 20..." (Common statement in media). Theory – Intro

40S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Statistics Lesson: ST-6 Confidence Intervals Confidence Intervals and the Standard Normal Curve All our confidence interval work is based on predictions from normal data. To calculate our confidence interval, all we need are the  and  parameters (the same parameters that we always work with when using normal data). The graph shows a standard normal curve with an interval that includes 95 percent of the data. The 95 percent interval is symmetric about the mean. This means that each unshaded tail of the graph represents 2.5 percent of the data. If we use IT to calculate the z-scores, we get: z 1 = z 2 = 1.96 prob above low tail confirmation

40S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Statistics Lesson: ST-6 Confidence Intervals Confidence Intervals and the Standard Normal Curve z 1 = z 2 = 1.96 Therefore, for a normal distribution with mean  and standard deviation , a 95 percent confidence interval can be defined as follows: 95% confidence interval =  ± 1.96  = (  ,   ) The interval represents the range of values that lie within 1.96 standard deviations of the mean. This means that the probability of any particular data value lying in the interval is 0.95 (To find a different type of confidence interval eg. 90%, 99%, we calculate the corresponding z-scores, and use them in place of the 1.96 in the formula above.) 90%: %: 2.575

40S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Statistics Lesson: ST-6 Confidence Intervals Confidence Interval for Cap and Gown Versus Formal Wear A survey of graduates includes 40 students. 17 prefer to dress formally, rather than use caps and gowns. Based on this data, what would be our prediction for the number of students in another group of 40 that would favor formal dress, expressed as a 95% confidence interval? First, we test for an approximation to a normal distribution: np = 17 > 5, nq = 23 > 5 Therefore, the normal approximation to the binomial is OK. For a normal approximation: To construct a 95% confidence interval for the number of students in favour of formal wear, you do the following: 95 % CI =  ± 1.96  = 17 ± 1.96(3.13) = 17 ± 6.13 = (10.87, 23.13) Therefore, in a sample of 40 students, we can be 95 percent confident that from 11 to 23 (inclusive) students prefer formal dress.

40S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Statistics Lesson: ST-6 Confidence Intervals Confidence Interval for Cap and Gown Versus Formal Wear (cont’d) Instead of stating our confidence interval using raw scores, we can use percentages. To construct a 95 percent confidence interval of the percent of students that prefer formal dress, you do the following (express each raw score as a percentage of the entire population): Therefore, in a sample of 40 students, we can be 95 percent confident that from 27 percent to 58 percent of the students prefer formal wear.

40S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Statistics Lesson: ST-6 Confidence Intervals Theory – Confidence Interval and Sample Size Prove that a normal approximation is okay

40S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Statistics Lesson: ST-6 Confidence Intervals Theory – Confidence Interval and Sample Size (cont’d)

40S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Statistics Lesson: ST-6 Confidence Intervals Theory – Confidence Interval and Sample Size (cont’d)

40S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Statistics Lesson: ST-6 Confidence Intervals Theory – Margin of Error, and Percent Margin of Error The question and the solution from the previous page are repeated here: In a survey of 96 students, 41 preferred formal wear. The 95 percent confidence interval for the number of students who prefer formal wear is as follows: 95% confidence interval =  ± 1.96  = 41 ± 1.96(4.85) = 41 ± 9.51 The margin of error is defined as half the confidence interval, from  to the lower or upper boundary of the confidence interval. That is, margin of error = 1.96  Margin of error = ± 1.96(4.85) = ± 9.51

40S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Statistics Lesson: ST-6 Confidence Intervals Theory – Margin of Error, and Percent Margin of Error (cont’d) The percent margin of error is defined as the margin of error expressed as a percent of the sample size. Therefore, in the above example, the percent margin of error is determined as follows:

40S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Statistics Lesson: ST-6 Confidence Intervals Example - Margin of Error, and Percent Margin of Error

40S Applied Math Mr. Knight – Killarney School Slide 13 Unit: Statistics Lesson: ST-6 Confidence Intervals Example - Margin of Error, and Percent Margin of Error (cont’d)

40S Applied Math Mr. Knight – Killarney School Slide 14 Unit: Statistics Lesson: ST-6 Confidence Intervals Example - Margin of Error, and Percent Margin of Error (cont’d)

40S Applied Math Mr. Knight – Killarney School Slide 15 Unit: Statistics Lesson: ST-6 Confidence Intervals Example - Sample Problem (2) Confidence Interval and Margin of Error

40S Applied Math Mr. Knight – Killarney School Slide 16 Unit: Statistics Lesson: ST-6 Confidence Intervals Example - Sample Problem (2) Confidence Interval and Margin of Error (cont’d)