There is a small table, entitled Common Critical Values, located at the bottom right corner of the Positive zscores table. A confidence level of 90%, 0.90 matches up with a critical value of 1.645. 1.645 is the 20:12 we use for a confidence level of 90%. In the same fashion, from the table, 1.96 is the 20:2 for 95% and 2.575 is the Zo/Z for 99%. For your ease of reference, I have recreated the 2 by 4 table. Common Critical Values Confidence Level Critical Value 0.90 1.645 0.95 1.96 0.99 2.575 Now we will put this all together into an example. Page 2 of 4 EXAMPLE 1 A sample of 850 people responded to a yeso survey question. 70% said yes. Not every member of the population was asked, so we must report our results with a margin of error. We will report the results at a confidence level of 95%. pa E = Za/2 1 n Za/2 for 95% is 1.96 p = 70% = 0.7 q = 30% = 0.3 n = 850 0.21 E = 1.96 0.7) (0.3) = 1.96 = 1.9610.000247059 = 350 850 1.96(0.0157181106) = 0.0308 = 0.031 = 3.1% The margin of error is + 3.1% p = 70%. 70% of the people we surveyed said yes. But we did not survey the entire population, so we must attach our calculated margin of error, E = 3.1%. We will report our answer in the form p + E. This form is called a confidence interval. Conclusion: At the 95% confidence level, we estimate the actual percentage of the population that would say yes is contained in the interval 70% + 3.1%. Restated, in the interval from 66.9% to 73.1%. Page 3 of 4EXAMPLE 2 A sample of 1,000 people were surveyed and 750, 75%, said yes. Construct a 90% confidence interval estimate of the population percentage (the percentage we estimate would say yes if we could ask the entire population). Zd/2 at 90% =1.645 ,0 = 75% = 0.75 q = 25% = 0.25 n = 1,000 E: 1.645 lw = 1.645 l0'1875 = 1.645(0013693) = 1,000 1000 = 0.0225 = 0.023 =2.3% Conclusion: At the 95% confidence level, we estimate the actual percentage of the population that would say yes is contained in the interval 75% 1r 2.3%. EXAMPLE 3 A sample of 960 people were surveyed and 28% said yes. Construct a 99% confidence interval estimate of the population percentage (the percentage we estimate would say yes if we could ask the entire population). Zd/Z at 99% = 2.575 p = 28% q = 72% n = 960 E = 2.575 lw = 0.0373 = 3.7% Conclusion: At the 99% confidence level, we estimate the actual percentage of the population that would say yes is contained in the interval 28% i 3.7%. Page 4 of 4 Quiz 1 Chapter 7 As always, show your work. Report your answers as confidence intervals in form, 13 i E like 28% 1 3.7% in the last example of the lesson. 1) A sample of 820 people were surveyed. 32% said yes. Construct a 90% confidence interval estimate for the percentage that would say yes, If the entire population could be surveyed. 2) A sample of 645 people were surveyed. 81% said yes. Construct a 95% confidence interval estimate for the percentage that would say yes, If the entire population could be surveyed. 3) A sample of 1,250 people were surveyed. 63% said yes. Construct a 99% confidence interval estimate for the percentage that would say yes, If the entire population could be surveyed. 4) A sample of 623 people were surveyed. 21% said yes. Construct a 95% confidence interval estimate for the percentage that would say yes, If the entire population could be surveyed