General Confidence Intervals Section

Starter A shipment of engine pistons are supposed to have diameters which vary according to N(4 in, 0.1 in) A sample of 10 pistons has an average diameter of 4.05 inches State a 95% confidence interval for the true mean diameter of all the pistons

Today’s Objectives Find the z* critical value associated with a level C confidence interval Find a confidence interval for any specified confidence level C –(In other words, let’s remove the need for the rule) California Standard 17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error

Confidence Intervals A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter. To get confidence level C we must catch the central probability C under a normal curve So we define a value called z* such that the area under a standard normal curve between –z* and +z* is C

Finding the z* values Suppose we want a 90% confidence interval Then 90% of the area under the curve must be between –z* and +z* Since the curve is symmetric, that means that 5% of the area is below –z* and 5% is above +z* –So how much area is below +z*? Search Table A for the z value that has 95% of the area to its left –z is between 1.64 and 1.65, so we can use Use the calculator to get the same result –invnorm(.95) = Use Table C to get the same result –z* values are in the bottom row above the values for C

The important z* values You have found the z* associated with a 90% C.I. Now find z* for a 95% C.I. and for a 99% C.I. Summarize your results in a simple table Confidence Level Z* 90% % %2.576

Using z* to form a C.I. The form of confidence intervals is estimate ± margin of error The margin of error is a number of standard deviations –In our example yesterday, we used 2 s.d. Since z* is measured in standard deviations, multiply by the s.d. of the sampling distribution to get margin of error Then add and subtract the margin to the estimate So here is the formula for forming a level C confidence interval:

Example 10.4 Repeated weighings of the active ingredient in a painkiller are known to vary normally with a standard deviation of.0068g Three specimens weigh: g0.8363g0.8447g Form a 99% confidence interval for the mean weight of the ingredient.

Step-By-Step Answer 1.Find the sample mean 2.Find the standard deviation of sample means 3.Use z* = in the formula to form the confidence interval 4.Conclusion: I am 99% confident that the true mean weight is between g and g

Example 10.4 Modified Repeated weighings of the active ingredient in a painkiller are known to vary normally with a standard deviation of.0068g Three specimens weigh: g0.8363g0.8447g Form a 95% and a 90% confidence interval for the mean weight of the ingredient.

Step-By-Step Answer: 95% 1.Find the sample mean 2.Find the standard deviation of sample means 3.Use z* = in the formula to form the confidence interval 4.Conclusion: I am 95% confident that the true mean weight is between g and g

Step-By-Step Answer: 90% 1.Find the sample mean 2.Find the standard deviation of sample means 3.Use z* = in the formula to form the confidence interval 4.Conclusion: I am 90% confident that the true mean weight is between g and g

Conclusion Describe the change in the confidence intervals we found as we changed C. As C decreased from 99% to 95% to 90% the intervals got narrower. –In other words, more accurate. What did we give up to get the increased accuracy? We reduced confidence. In the last case, we used a method that gives correct results in 90% of all samples, not 99%.

Today’s Objectives Find the z* critical value associated with a level C confidence interval Find a confidence interval for any specified confidence level C –(In other words, let’s remove the need for the rule) California Standard 17.0 Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error

Homework Read pages Do problems 5, 7, 8