Chapter 8: Estimation Confidence Intervals

Sec. 8.1 Basics of Confidence Intervals Point estimator: the sample mean or sample proportion. 𝑥 or 𝑝 We want to know how good of an estimate this is. A confidence interval is the point estimator plus or minus a margin of error, denoted 𝜃 ±𝐸

Interpretation of C.I.s Example: 𝑝 = .70 Your confidence interval is .70 ± .04, or 0.66<𝑝<0.74, or 0.66 to 0.74 α = 0.05 Interpretation: “There is a 95% chance that the interval 0.66 to 0.74 contains the true population proportion.” Misinterpretation: “There is a 95% chance that the true value of 𝑝 will fall between 0.66 and 0.74”

Affect of Confidence Level on Width There’s a trade-off between width and confidence level. You can be really confident about your answer but your answer will not be very precise. Or you can have a precise answer (small margin of error) but not be very confident about your answer.

Affect of Sample Size on Width A larger sample size makes the width of the interval narrower. Large samples are closer to the true population so the point estimate is pretty close to the true value.

Examples Suppose you have a confidence interval with a sample size of 30. What will happen to the confidence interval if the sample size decreases to 25? Suppose you compute a 90% confidence interval. What will happen to the confidence interval if you increase the confidence level to 95%?

Examples A study from a random sample of Krispy Kream donuts found a 95% confidence interval of 7 < µ < 13, where µ is the grams of sugar in a donut. State the statistical interpretation. “There is a 95% chance that the interval 7 to 13 contains the true mean amount of sugar in a Krispy Kream donut.”

Sec. 8.2 One-Sample Interval for the Proportion 𝑝 ± 𝑍 𝐶 𝑝 (1− 𝑝 ) 𝑛 𝑝 = sample proportion 𝑧 𝑐 = critical value 𝑛= number of sample values

Critical Value (Zc) The critical value is a value from the normal distribution. You can use the invNorm on the calculator to find the critical value. For 95%: invNorm(.975) = 1.96 (Why .975?)

Standard error One important part of the calculation was 𝑝 (1− 𝑝 ) 𝑛 This is the Standard Error. The confidence interval is always a point estimate plus or minus a Margin of Error. The Margin of Error is always a critical value times a Standard Error.

Example A concern was raised in Australia that the percentage of deaths of Aboriginal prisoners was higher than the percent of deaths of non-Aboriginal prisoners, which is 0.27%. A sample of six years (1990-1995) of data was collected, and it was found that out of 14,495 Aboriginal prisoners, 51 died ("Indigenous deaths in," 1996). Find a 95% confidence interval for the proportion of Aboriginal prisoners who died.

Example 𝑝 ± 𝑍 𝐶 𝑝 (1− 𝑝 ) 𝑛 𝑝 = 51/14495 = .003518 𝑧 𝑐 = 1.96 𝑛= 14495 .003518±1.96 .003518(1−.003518) 14495 .002554 < p <.004482 We are 95% confident that the true proportion of Aboriginal prisoners who dies is between .00255 and .004482.

Example with a calculator A researcher is studying the effects of income levels on breastfeeding of infants hypothesizes that countries where the income level is lower have a higher rate of infant breastfeeding than higher income countries. It is known that in Germany, considered a high-income country by the World Bank, 22% of all babies are breastfeed. In Tajikistan, considered a low-income country by the World Bank, researchers found that in a random sample of 500 new mothers that 125 were breastfeeding their infants. Find a 90% confidence interval of the proportion of mothers in low-income countries who breastfeed their infants?

We are 90% confident that the true proportion of women in low-income countries who breastfeed their infants is between 21.8% and 28.2%

Sec. 8.3 One-Sample Interval for the Mean Confidence interval for a mean Population standard deviation is not known 𝑥 ± 𝑡 𝑐 𝑠 𝑛 𝑥 is the point estimator for  𝑡 𝑐 is the critical value 𝑠 is the sample standard deviation 𝑛 is the sample size

How to Find 𝑡 𝑐 You can use a chart (A.2) in your book if you’d like On the ti, go to STAT →TESTS →T-TINTERVAL… Enter 𝑥 =0, 𝑆 𝑥 = 𝑛 , and the other information as given The greater number in the interval is the value for 𝑡 𝑐

IQs of the famous A random sample of 20 IQ scores of famous people was taken information from the website of IQ of Famous People and then using a random number generator to pick 20 of them. Find a 98% confidence interval for the IQ of a famous person. 158 180 150 137 109 225 122 138 145 118 126 140 165 170 105 154

IQs of the famous 𝑥 ± 𝑡 𝑐 𝑠 𝑛 𝑥 = 145.4 𝑡 𝑐 = 2.539 𝑠 = 29.27 𝑛 = 20 𝑥 ± 𝑡 𝑐 𝑠 𝑛 𝑥 = 145.4 𝑡 𝑐 = 2.539 𝑠 = 29.27 𝑛 = 20 145.4±2.539 29.27 20 =145.4±16.6 (128.8, 162) There is a 98% chance that the mean IQ score of a famous person is between 128.8 and 162.

Male Life Expectancy in Europe The data in table #8.3.3 are the life expectancies for men in European countries in 2011 ("WHO life expectancy," 2013). Find the 99% confident interval for the mean life expectancy of men in Europe 73 79 67 78 69 66 74 71 75 77 68 81 80 62 65 72 70 63 82 60  

Male Life Expectancy in Europe The data in table #8.3.3 are the life expectancies for men in European countries in 2011 ("WHO life expectancy," 2013). Find the 99% confident interval for the mean life expectancy of men in Europe Using the calculator (or excel), we find 𝑥 =73.73585 and 𝑆 𝑥 =5.728307 Counting the number of data, we have 𝑛=53

TInterval on TI

ZInterval The book does not cover it, but we will also use ZInterval on the calculator. We use this when we want a confidence interval about a population mean and we know the population standard deviation. The steps are exactly like the Tinteveral, except Zinterval will ask for 𝜎 as well.