What Does “95% Confident” Mean?

University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

462

Pages:

2
Author:

camisants7m6n

What does “95% confident” mean? In this Activity, we will use the Confidence Intervals applet to simulate the process of taking an election poll. Some proportion p of the population plans to vote for our candidate. If we sample n people from the population, the proportion of people in the sample who say they plan to vote for our candidate is our best estimate of the population proportion. 1. Set the population proportion to 0.55, the confidence level to 95% and the sample size to 75. 2. Click “Sample” to choose an SRS and display the resulting confidence interval. The confidence interval is displayed as a horizontal line segment with a dot representing the sample proportion in the middle of the interval. The true proportion (p) is the green vertical line. Interpret the confidence interval: I am 95% confident that the true proportion of people who plan to vote for my candidate is between 46% and 68%. Did the first confidence interval capture the true proportion? Yes, it captures the green line. Repeat this 10 times and sketch what you see to the right. How many of the intervals capture the true proportion? All intervals capture the true proportion. 3. “Reset” and then take a total of 100 confidence intervals (sample 25 four times). How many out of 100 captured the true proportion? Is this surprising? Why? Answers will vary. Around 95. Not surprising, because it corresponds to the confidence level of 95%. 4. Watch your confidence intervals as you drag the confidence level from 95% to 99% (don’t “Reset). What happens to the intervals when the confidence level is increased? Why does this make sense? The interval gets wider. It makes sense because the confidence level is higher. Higher confidence – wider interval (and larger margin of error) 5. “Reset”, then sample 100 times at an 80% confidence level. What percent of the intervals capture the true proportion? Answers will vary. Around 80%. Interpret the confidence level: If we sample many times and make 80% confidence intervals, about 80% of them will capture the true proportion. 6. Now we will see what happens when we adjust the sample size. Change the sample to 20 and sample for 1 interval. Then change it to 250 and sample for 1 interval. What happens to the interval when the sample size is increased? Why? The interval gets wider for n=20, and narrower for n=250. We get more precise results with larger sample. 7. We want to estimate the proportion of people who will vote for our candidate within 5% margin of error and with 95% confidence. How many people should be in our sample? 𝑝𝑞 𝑛 𝑀𝐸 = 𝑧 ∗ √ 0.05 = 1.96√ 𝑀𝐸 = 0.05 0.573 ∙ 0.427 𝑛 0.05 0.573 ∙ 0.427 =√ 1.96 𝑛 𝑝 = 0.573 𝑞 = 0.427 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 1.96 𝑠𝑞𝑢𝑎𝑟𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 (0.05) 2 0.573 ∙ 0.427 = (1.96)2 𝑛 𝑛= 𝑓𝑜𝑟 95% 𝐶𝐼 𝑧 ∗ = 1.96 0.573 ∙ 0.427 ∙ (1.96)2 = 375.97 (0.05)2 𝑐𝑟𝑜𝑠𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑎𝑙𝑤𝑎𝑦𝑠 𝑟𝑜𝑢𝑛𝑑 𝑢𝑝! 𝑛 = 376 − 𝑝𝑒𝑜𝑝𝑙𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 8. We want to estimate the proportion of people who will vote for our candidate within 2% margin of error and with 98% confidence. How many people should be in our sample? 𝑝𝑞 𝑛 𝑀𝐸 = 𝑧 ∗ √ 0.02 = 2.33√ 𝑛= 𝑀𝐸 = 0.02 0.573 ∙ 0.427 𝑛 0.573 ∙ 0.427 ∙ (2.33)2 = 3320.7 (0.02)2 𝑓𝑜𝑟 95% 𝐶𝐼 𝑧 ∗ = 2.33 𝑝 = 0.573 𝑞 = 0.427 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑛 𝑎𝑙𝑤𝑎𝑦𝑠 𝑟𝑜𝑢𝑛𝑑 𝑢𝑝! 𝑛 = 3321 − 𝑝𝑒𝑜𝑝𝑙𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒