Estimating and Constructing Confidence Intervals

10.1 Interval Estimation Finding a value for or 95% 99%

10.2 Confidence Intervals for the Mean with Known Population Variance The population mean is an unknown parameter is a statistic which estimates We call a point estimate because its value is a point on the real number line Unfortunately, if we sample from a continuous distribution,

Statisticians prefer interval estimates E (the Error Tolerance) depends on the sample size, how certain we want to be that we are correct (Level of Confidence), and the amount of variability in the data When is known has approximately a standard normal distribution.

Therefore,

Notice that increasing the level of confidence, decreases the probability of error, however it also increases E creating a wider interval Notice as sample size increases, E decreases creating a more narrow interval Notice the more variability in the population, the larger E creating a wider interval

Example A sample of 100 visa accounts were studied for the amount of unpaid balance. Construct and interpret a 95% confidence interval We are 95% confident that the mean unpaid balance of visa accounts is between $ and $ Construct a 99% confidence interval We are 99% confident that the mean unpaid balance of visa accounts is between $ and $

Choosing the Sample Size In the design stages of statistical research, it is good to decide in advance the confidence level you wish to use and to select the error tolerance you want for the project. This will let us decide how big our sample needs to be. Sample size for estimating If the value of is not known, we do preliminary sampling to approximate it.

Example We wish to estimate the number of patient-visit hours per week physicians in solo practice spent. How large a sample is needed if we want to be 99% confident that our point estimate is within 1 hour of the population mean? Assume a standard deviation of hours.

10.3 Student’s t Distribution is unknown To avoid the error involved in replacing by, we will introduce a new random variable called Student’s t variable. (t-distribution) If we sample from a normal distribution has a t-distribution with n-1 degrees of freedom.

Properties of the t-distribution continuous and symmetric about 0 more variable and slightly different shape than the standard normal As n becomes large, the t distribution can be approximated by the standard normal distribution (The bottom row of the t-distribution is Z) With a sample size of 11 a Confidence level of 95%, what is the two tailed t value?

10.4 Confidence Intervals for the Mean with Unknown Population Variance For unknown,

Example Mileage of tires in 1000’s of miles Sample: 42, 36, 46, 43, 41, 35, 43, 45, 40, 39 Compute a 95% confidence interval for We are 95% confident that the population mean mileage of tires is between 38,432 and 43,568 miles.

Example A random sample of 20 apples yields and Find a 99% confidence interval. We are 99% confident that the population mean weight of apples is between and oz.

10.5 Confidence Intervals for Proportions (Large Samples) For large n, is approximately standard normal.

Example A survey of 1,200 registered voters yields 540 who plan to vote for the republican candidate. p = proportion of all voters who plan to vote for the republican candidate Find a 95% confidence interval for p We are 95% confident that the population proportion of voters who plan to vote for the republican candidate is between 42.2% and 47.8%.