Confidence Intervals II

Confidence Intervals II Chapter 18 Which is true about a 95% confidence interval based on a given sample? I. The interval contains 95% of the population. II. 95% of our samples will contain the true proportion. III. The interval is narrower than a 98% confidence interval would be. A) None B) I only C) II only D) III only E) II and III only o When we make an interval, our goal is to accurately estimate the unknown p. o Interval is like a net: the bigger the net (the interval), the more likely we to catch the true value of p. o More confidence = bigger interval (and bigger margin of error) o Confidence interval is a balance between certainty and precision Margin of Error MOE = z*
To make margin of error smaller: o Reduce level of confidence o Increase sample size   MOE gets smaller when n gets larger. Because we take the square root of n we must take four times as many observations in order to cut the margin of error in half. We could choose a sample size to get desired margin of error: ∗ = ̂ ( )  * ̂ = 0.5 if no info is given; it makes ̂  largest. When you don’t have a ‘best guess estimate’ for your proportion of successes in the population, we make ̂ = .5 (because that’s our most conservative estimate of probability for success: 50/50). Example 1 Suppose a 90% confidence interval is stated as (0.30, 0.42). What is the sample proportion from this sample? 0.36 – right in the middle (or (0.42-0.30)/2=0.06 0.30+0.06=0.36) What is the margin of error? 0.06 ← 90% 0.30 0.36±0.06 → 0.42 Example 2 A teacher finds that the proportion of student coming to class without homework completed is increasing as the weather gets nicer. She finds a 95% interval for the proportion of students who have not completed homework to be (0.15, 0.25). What is the margin of error? a) 0.2 b) 0.05 ← c) 0.1 0.15 95% → 0.25 d) 0.4 e) We cannot tell without numbers to plug into the calculator. Example 3 How large of a sample do I need to estimate the proportion of students who have seen “Inside Out” at 95% confidence with a margin of error of 5%? ∗  = 0.05 = 1.96 ̂ = 0.5  = 0.5 1.96  = 0.5 ∗ 0.5 ∗   = 384.16 0.05 Always round up!!! We will need 385 students in our sample Example 4 How large of a sample do I need to estimate the proportion of CA students who have seen “Inside Out” if 33% of my class has seen it? I want 90% confidence and 3% margin of error.  = 0.03 ∗ = 1.645 ̂ = 0.33  = 0.67 1.645  = 0.33 ∗ 0.67 ∗   = 664.78 − !!" 665 #$%"! $# 0.03 Example 5 a) An experiment finds that 27% of 53 subjects report improvement after using a new medicine. Create a 95% confidence interval for the actual cure rate. ̂ = 0.27  = 0.73 = 53 0.27 ∗ 0.73 ' = ( = 0.061 53 0.27 ± 1.96 ∗ 0.061 (0.1504,0.3896) Why is this interval so wide? Because our confidence level is high. b) Make it narrower – 90% confidence. 0.27 ± 1.645 ∗ 0.061 (0.1697,0.3703) What are the advantages and disadvantages? The interval is narrower, but our confidence level is lower. c) What sample size would we need in a follow-up study if we want a margin of error of 5% with 98% confidence?  = 0.05 ∗ = 1.96 ̂ = 0.27  = 0.73 2.33  = 0.27 ∗ 0.73 ∗   = 428.01 0.05 Example 6 A newspaper reports that the governor’s approval rating stands at 65%. The article adds that the poll is based on a random sample of 972 adults and has a margin of error of 2.5%. What level of confidence did the pollsters use?  = 0.025 ̂ = 0.65  = 0.35  972 = 0.65 ∗ 0.35 ∗ + , 0.025 ∗ ∗ = 2.67 = 1.63 90% Confidence Level = 972