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Confidence Intervals Day 2 Last class we saw "what" a confidence interval is. We know that an 80% confidence level means that 80 out of 100 samples will produce an interval that contains the actual parameter we are interested in finding. We also saw that all confidence intervals have the basic form: estimate±marginoferror We can rewrite this as the following: estimate±(criticalvalue)(standarddeviationofstatistic) or Example 1: zc can be found using our z-score table. a) Find zc when c = 80 b) Find zc when c = 95 A quick reference for common zc Level of Confidence zc 80 ____________ 90 1.645 95 _____________ 99 2.575 Remember using table A to find critical values (z-scores) only works if x-bar is NORMAL! Don't forget about the CLT. Example 2: Myface, a social networking site, allows its users to add friends, send messages, and update their personal profiles. The following represents a random sample of the number of friends for 40 users of the website. Assume that the standard deviation is σ = 57 users. 140 105 130 97 80 165 232 110 214 201 122 98 65 88 154 133 121 82 130 211 153 114 58 77 51 247 236 109 126 132 125 149 122 74 59 218 192 90 117 105 a) Calculate a point estimate for the number of friends a Myface user has on average. b) Use a 95% confidence level to find the margin error for the mean number of friends of all users of the myface website. c) Construct a confidence interval using part (b) and (c) d) Interpret what the confidence interval means. e) Create confidence interval using a 99% confidence level f) Compare the widths of part (e) and (c) Example 3: A random sample of 34 home theater systems has a mean price of $452.80. The standard deviation for all home theater systems is $85.50. Construct a 94% confidence interval and interpret the results. Example 4: A random sample of 45 soccer coaches found the mean number of years coaching to be 6.8 years. The standard deviation for all soccer coaches is 4.7 years. Construct a 90% confidence interval and interpret the results. Calculating a Required Sample Size By using the margin of error formula, we can calculate the required sample size to reach a certain margin of error. You will often have an idea of the highest error that you will want and will be okay with a smaller margin of error so we'll write it as an inequality Example 5: Researchers would like to estimate the mean cholesterol level, μ, of a particular variety of monkey that is often used in laboratory experiments. They would like their estimate to be within 1 milligram per deciliter (mg/dl) of the true value of μ at a 95% confidence level. A previous study involving this variety of monkey suggests that the standard deviation of cholesterol level is about 5 mg/dl. Obtaining monkeys is time consuming and expensive, so the researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate. Example 6: A restaurant owner wishes to find the 92% confidence interval of the true mean cost of a meal for a costumer. How large should the sample be if she wishes to be accurate within $0.10? A previous study showed that the standard deviation of the price was $1.25. Example 7: You want to estimate the mean number of friends for all users on Myface. How many users must be included in the sample if you want to be 98% confident that the sample mean is within 7 friends of the population mean? (remember we were given the standard deviation in example 2)