Section 10.1 Confidence Intervals The Basics

Confidence Intervals! A level C confidence level 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. A confidence interval is found by the following formula: estimate ± margin of error The margin of error shows how accurate we believe our guess is, based on the variability of the estimate.

Confidence Intervals! Any confidence interval has two parts: An interval computed from the data A confidence level giving the probability that the method produces an interval that covers the parameter.

Critical Values! The number z* with probability p lying to its right under the standard normal curve is called the upper p critical value of the standard normal distribution. * Make sure you know how to read the table of critical values.

Conditions for Inference about a Population Mean Random--The data is an SRS from the population or from a randomized experiment Observations from the population have a normal distribution with an unknown mean (μ) and standard deviation (σ) Independence is assumed for the individual observations when calculating a confidence interval. When we are sampling without replacement from a finite population, it is sufficient to verify that the population is at least 10 times the sample size in order to use our st. dev formula.

CAUTION Be sure to check that the conditions for constructing a confidence interval for the population mean are satisfied before you perform any calculations.

Here’s how to find the level C confidence interval! Any normal curve has probability C between point z* standard deviations below its mean and the point z* standard deviations above its mean. The standard deviation of the sampling distribution of is /√n and its mean is the population mean . So there is probability C that the observed sample mean is between  - z*(/√n) and  + z*(/√n) Whenever this happens, the population mean  is contained between - z*(/√n) and + z*(/√n) This is the confidence interval. The estimate of the unknown  is , and the margin of error is z*(/√n)

Possible Statements About a Confidence Interval: “I am __% confident that the true (parameter) is between __ and __ because when I computed the endpoints of the interval I followed a procedure that creates intervals containing the true (parameter) about __% of the time it is followed.” “This __% confidence interval is one result of a method that had a __% chance of producing an interval capturing the true (parameter).” “This __% confidence interval is one observed result of a process that, were it to be repeated over many random samples, would produce intervals containing the population parameter __% of the time.” “I am 100% confident that the sampling procedure I used has a __% chance of obtaining a sample with a (statistic) such that the interval constructed around the (statistic) contains the (parameter).”

Coach Adams’ Standard Comment KEEP IN MIND, I change how I say things, but I never change the fact that I must comment on what I am confident about, why I have that level of confidence, and INCLUDE CONTEXT! Since we don’t want to forget context, imagine that when I conducted a 99% confidence interval on the Capture the Mean Activity that I got (76.7, 91.7). Here is what I might say: “I am 99% confident that the actual course average for students in the Fall 2009 AP Statistics class was between a 76.7 and a 91.7 because the methods I used results in intervals such that 99% of all intervals created will capture this true average grade.”

Helping Us Understand Confidence Intervals The confidence interval mentioned on the previous slide (76.7, 91.7) was calculated with 99% confidence. Essentially, this particular interval is based on one sample so it may be one of the 99% of all intervals that captures  or it may be one of the 1% of all intervals that fails to capture . The confidence level is the probability that the METHOD gives an interval that captures the true parameter.

Our “confidence” is in the PROCEDURE used to generate the interval. REMEMBER Our “confidence” is in the PROCEDURE used to generate the interval.

INFERENCE TOOLBOX (p 631) Steps for constructing a CONFIDENCE INTERVAL: 1—PARAMETER—Identify the population of interest and the parameter you want to draw a conclusion about. 2—CONDITIONS—Choose the appropriate inference procedure. VERIFY conditions (Random, Normal, Independent) before using it. 3—CALCULATIONS—If the conditions are met, carry out the inference procedure. 4—INTERPRETATION—Interpret your results in the context of the problem. CONCLUSION, CONNECTION, CONTEXT (meaning that our conclusion about the parameter connects to our work in part 3 and includes appropriate context)

What does a confidence interval look like? There are two commonly used formats for expressing a confidence interval Estimate ± margin of error Example: 57.39 ± 3.97 Interval notation Example: (53.42, 61.36)

How confidence intervals behave! Margin of error gets smaller when z* gets smaller. Smaller z* is the same as smaller confidence level C. To obtain a smaller margin of error, you must be willing to accept lower confidence.  gets smaller. The standard deviation  measures the variation in the population. It is easier to pin down  when  is smaller. n gets larger. Increasing the sample size n reduces the margin of error for any fixed confidence interval. Because n appears under the square root sign, we must take four times as many observations in order to cut the margin of error in half.

Sample size for desired margin of error! Ultimately, the population standard deviation is being given to us, so this doesn’t change. And we are typically given a set confidence level, C, which directly determines z*. So, ultimately, we can only change our sample size to achieve the desired SMALL INTERVAL WITH A HIGH LEVEL OF CONFIDENCE! To determine the sample size n that will yield a confidence interval for a population mean with a specific margin of error m, set the expression for the margin of error to be less than or equal to m and solve for n: ALWAYS ROUND UP to the next whole number value of n to ensure that your sample is large enough to give desired margin of error (or smaller).

Sample Size Example Again, consider our Capture the Mean activity. If we want to estimate the true mean within 5 points, how big of a sample do we need? σ=6.4551; 99% confidentz*=2.576 So we need at least 12 grades in our sample to bring the margin of error to within 5 points. (Notice that we rounded 11.06 up to 12.)

Some cautions! The data must be an SRS sample from the population (or from a randomized experiment). The formula is not correct for probability sampling designs more complex than an SRS. There is no correct method for inference from data haphazardly collected with bias of unknown size. Because is strongly influenced by a few extreme observations, outliers can have a large effect on the confidence interval. Search for outliers and try to correct them or justify their removal before computing the interval. If the sample size is small and the population is not normal, the true confidence interval will be different from the value C used in computing the interval. If n≥ 15 then the confidence interval will not be greatly disturbed by nonnormal populations. You must know the standard deviation of the population.

Technology Disclaimer As always, you will be allowed unrestricted use of your calculator on quizzes and tests (as well as the actual AP Exam). For this reason, ALWAYS be certain to write down the values of key numbers that are being used (means, standard deviations, degrees of freedom, significance levels, etc.) along with results of the calculator procedures in order to receive full credit. To use your calculator: STAT: TESTS: Z Interval Make sure you are comfortable with difference between when you have data versus stats Plug in exactly what you are asked for.