FINAL EXAMINATION STUDY MATERIAL II Chapter 19 Parts Of Chapter 23 CONFIDENCE INTERVALS FOR ONE PROPORTION CONFIDENCE INTERVALS FOR ONE MEAN

Point Estimate and Interval Estimate A point estimate is a single number that is our “best guess” for the parameter. Point estimation produces a number (an estimate) which is believed to be close to the value of the unknown parameter. An interval estimate is an interval of numbers within which the parameter value is believed to fall. Interval estimation produces an interval that contains the estimated parameter with a prescribed confidence.

Point Estimate and Interval Estimate (Figure 1)

Point Estimate and Interval Estimate Figure 1: A point estimate predicts a parameter by a single number. An interval estimate is an interval of numbers that are believable values for the parameter. Question: Why is a point estimate alone not sufficiently informative?

Point Estimate and Interval Estimate A point estimate doesn’t tell us how close the estimate is likely to be to the parameter. An interval estimate is more useful, it incorporates a margin of error which helps us to gauge the accuracy of the point estimate.

The Logic behind Constructing a Confidence Interval To construct a confidence interval for a population proportion, start with the sampling distribution of a sample proportion. The sampling distribution: (1) Is approximately a normal distribution for large random samples by the CLT. (2) Has mean equal to the population proportion. (3) Has standard deviation called the standard error.

Confidence Interval or Interval Estimate Sample estimate  Multiplier × Standard Error Sample estimate  Margin of error Multiplier is a number based on the confidence level desired and determined from the standard normal distribution (for proportions) or Student’s t-distribution (for means).

The Multiplier Multiplier, denoted as z*, is the standardized score such that the area between -z* and z* under the standard normal curve corresponds to the desired confidence level. Note: Increase confidence level Implies larger multiplier

The Multiplier

For 90% Confidence Level

SOME CRITICAL VALUES FOR STANDARD NORMAL DISTRIBUTION C % CONFIDENCE LEVEL CRITICAL VALUE 80% 1.282 90% 1.645 95% 1.960 98% 2.326 99% 2.576

WHAT DOES C% CONFIDENCE REALLY MEAN? FORMALLY, WHAT WE MEAN IS THAT C% OF SAMPLES OF THIS SIZE WILL PRODUCE CONFIDENCE INTERVALS THAT CAPTURE THE TRUE PROPORTION. C% CONFIDENCE MEANS THAT ON AVERAGE, IN C OUT OF 100 ESTIMATIONS, THE INTERVAL WILL CONTAIN THE TRUE ESTIMATED PARAMETER. E.G. A 95% CONFIDENCE MEANS THAT ON THE AVERAGE, IN 95 OUT OF 100 ESTIMATIONS, THE INTERVAL WILL CONTAIN THE TRUE ESTIMATED PARAMETER.

CONFIDENCE INTERVAL FOR PROPORTION P [ONE-PROPORTION Z-INTERVAL] ASSUMPTIONS AND CONDITIONS RANDOMIZATION CONDITION 10% CONDITION SAMPLE SIZE ASSUMPTION OR SUCCESS/FAILURE CONDITION INDEPENDENCE ASSUMPTION NOTE: PROPER RANDOMIZATION CAN HELP ENSURE INDEPENDENCE.

Compact Formula For a Confidence Interval For a Population Proportion p is the sample proportion; z* denotes the multiplier; and is the standard error of .

Constructing a Confidence Interval to Estimate a Population Proportion The exact standard deviation of a sample proportion equals: This formula depends on the unknown population proportion, p. In practice, we don’t know p, and we need to estimate the standard error as

SAMPLE SIZE NEEDED TO PRODUCE A CONFIDENCE INTERVAL WITH A GIVEN MARGIN OF ERROR, ME SOLVING FOR n GIVES WHERE IS A REASONABLE GUESS. IF WE CANNOT MAKE A GUESS, WE TAKE

EXAMPLE 1 DIRECT MAIL ADVERTISERS SEND SOLICITATIONS (a.k.a. “junk mail”) TO THOUSANDS OF POTENTIAL CUSTOMERS IN THE HOPE THAT SOME WILL BUY THE COMPANY’S PRODUCT. THE RESPONSE RATE IS USUALLY QUITE LOW. SUPPOSE A COMPANY WANTS TO TEST THE RESPONSE TO A NEW FLYER, AND SENDS IT TO 1000 PEOPLE RANDOMLY SELECTED FROM THEIR MAILING LIST OF OVER 200,000 PEOPLE. THEY GET ORDERS FROM 123 OF THE RECIPIENTS. CREATE A 90% CONFIDENCE INTERVAL FOR THE PERCENTAGE OF PEOPLE THE COMPANY CONTACTS WHO MAY BUY SOMETHING. EXPLAIN WHAT THIS INTERVAL MEANS. EXPLAIN WHAT “90% CONFIDENCE” MEANS. THE COMPANY MUST DECIDE WHETHER TO NOW DO A MASS MAILING. THE MAILING WON’T BE COST-EFFECTIVE UNLESS IT PRODUCES AT LEAST A 5% RETURN. WHAT DOES YOUR CONFIDENCE INTERVAL SUGGEST? EXPLAIN.

SOLUTION

Solution Steps Using Technology TI – 83 PLUS COMMANDS Press STAT Scroll to TESTS Scroll to 1 - PropZInt Press ENTER Enter values for x; sample size, n; and confidence level

EXAMPLE 2 IN 1998 A SAN DIEGO REPRODUCTIVE CLINIC REPORTED 49 BIRTHS TO 207 WOMEN UNDER THE AGE OF 40 WHO HAD PREVIOUSLY BEEN UNABLE TO CONCEIVE. FIND A 90% CONFIDENCE INTERVAL FOR THE SUCCESS RATE AT THIS CLINIC. INTERPRET YOUR INTERVAL IN THIS CONTEXT. EXPLAIN WHAT “90 CONFIDENCE” MEANS. WOULD IT BE MISLEADING FOR THE CLINIC TO ADVERTISE A 25% SUCCESS RATE? EXPLAIN. THE CLINIC WANTS TO CUT THE STATED MARGIN OF ERROR IN HALF. HOW MANY PATIENTS’ RESULTS MUST BE USED? DO YOU HAVE ANY CONCERNS ABOUT THIS SAMPLE? EXPLAIN.

SOLUTION

EXAMPLE 3 A MAY 2002 GALLUP POLL FOUND THAT ONLY 8% OF A RANDOM SAMPLE OF 1012 ADULTS APPROVED OF ATTEMPTS TO CLONE A HUMAN. FIND THE MARGIN OF ERROR FOR THIS POLL IF WE WANT 95% CONFIDENCE IN OUR ESTIMATE OF THE PERCENT OF AMERICAN ADULTS WHO APPROVE OF CLONING HUMANS. EXPLAIN WHAT THAT MARGIN OF ERROR MEANS. IF WE ONLY NEED TO BE 90% CONFIDENT, WILL THE MARGIN OF ERROR BE LARGER OR SMALLER? EXPLAIN. FIND THAT MARGIN OF ERROR. IN GENERAL, IF ALL OTHER ASPECTS OF THE SITUATION REMAIN THE SAME, WOULD SMALLER SAMPLES PRODUCE SMALLER OR LARGER MARGINS OF ERROR?

SOLUTION

Effects of Confidence Level and Sample Size on Margin of Error The margin of error for a confidence interval: (i) Increases as the confidence level increases; (ii) Decreases as the sample size increases. For instance, a 99% confidence interval is wider than a 95% confidence interval, and a confidence interval with 200 observations is narrower than one with 100 observations at the same confidence level. These properties apply to all confidence intervals, not just the one for the population proportion.

What is the Error Probability for the Confidence Interval Method?

Inference About One Population Mean ESTIMATING MEANS WITH CONFIDENCE

Confidence Intervals for One Population Mean For large n, from any population, and also, For small n, from an underlying population that is normal; and the population standard deviation, is known; The confidence interval for the population mean is:

Example Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.75. Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85.

Solution

Solution Steps Using Technology TI – 83 PLUS COMMANDS Press STAT Scroll to TESTS Scroll to ZInterval Press ENTER Scroll to Stats Enter values for (i) the sample mean; (ii) the sample size; (iii) the standard deviation; (iv) the confidence level. Scroll to CALCULATE and press ENTER

Case II The sample size n is small; The population standard deviation is unknown;

Confidence Intervals for One Population Mean In practice, we don’t know the population standard deviation . Substituting the sample standard deviation s for to get introduces extra error. To account for this increased error, we must replace the z-score by a slightly larger score, called a t –score. The confidence interval is then a bit wider. This distribution is called the t distribution.

Summary: Properties of the t-Distribution The t-distribution is bell shaped and symmetric about 0. The probabilities depend on the degrees of freedom, . The t-distribution has thicker tails than the standard normal distribution, i.e., it is more spread out. A t -score multiplied by the standard error gives the margin of error for a confidence interval for the mean.

t - Distribution

t - Distribution The t Distribution Relative to the Standard Normal Distribution: The t distribution gets closer to the standard normal as the degrees of freedom ( df ) increase. The two are practically identical when . Question: Can you find z -scores (such as 1.96) for a normal distribution on the t table?

t - Distribution

t – Distribution Part of t - Table Displaying t-Scores. The scores have right-tail probabilities of 0.100, 0.050, 0.025, 0.010, 0.005, and 0.001. When and is the t -score with right-tail probability = 0.025 and two-tail probability = 0.05. It is used in a 95% confidence interval,

t - Distribution

t - Distribution The t Distribution with df = 6. 95% of the distribution falls between -2.447 and 2.447. These t -scores are used with a 95% confidence interval when n = 7. Question: Which t -scores with df = 6 contain the middle 99% of a t distribution (for a 99% confidence interval)?

Using the t Distribution to Construct a Confidence Interval for a Mean Summary: 95% Confidence Interval for a Population Mean When the standard deviation of the population is unknown, a 95% confidence interval for the population mean m is: To use this method, you need: Data obtained by randomization An approximately normal population distribution

ASSUMPTIONS AND CONDITIONS INDEPENDENCE ASSUMPTION: THE DATA VALUES SHOULD BE INDEPENDENT. THERE’S REALLY NO WAY TO CHECK INDEPENDENCE OF THE DATA BY LOOKING AT THE SAMPLE, BUT WE SHOULD THINK ABOUT WHETHER THE ASSUMPTION IS REASONABLE. RANDOMIZATION CONDITION: THE DATA SHOULD ARISE FROM A RANDOM SAMPLE OR SUITABLY A RANDOMIZED EXPERIMENT.

ASSUMPTIONS AND CONDITIONS 10% CONDITION: THE SAMPLE IS NO MORE THAN 10% OF THE POPULATION. NORMAL POPULATION ASSUMPTION OR NEARLY NORMAL CONDITION: THE DATA COME FROM A DISTRIBUTION THAT IS UNIMODAL AND SYMMETRIC. REMARK: CHECK THIS CONDITION BY MAKING A HISTOGRAM OR NORMAL PROBABILITY PLOT.

FINDING CRITICAL t - VALUES Using t tables (Table T) and/or calculator, find or estimate the 1. critical value t7* for 90% confidence level if number of degrees of freedom is 7 2. one tail probability if t = 2.56 and number of degrees of freedom is 7 3. two tail probability if t = 2.56 and number of degrees of freedom is 7 NOTE: If t has a Student's t-distribution with degrees of freedom, df, then TI-83 function tcdf(a,b,df) , computes the area under the t-curve and between a and b.

EXAMPLES

Other Factors That Affect the Choice of the Sample Size The first is the desired precision, as measured by the margin of error, m. The second is the confidence level. The third factor is the variability in the data. The fourth factor is cost.

What if You Have to Use a Small n? The t methods for a mean are valid for any n. However, you need to be extra cautious to look for extreme outliers or great departures from the normal population assumption. In the case of the confidence interval for a population proportion, the method works poorly for small samples because the CLT no longer holds.

More Examples From Practice Sheet

Solution Steps Using Technology TI – 83 PLUS COMMANDS Press STAT Scroll to TESTS Scroll to TInterval Press ENTER Scroll to Stats Enter values for (i) the sample mean; (ii) the sample size; (iii) the standard deviation; (iv) the confidence level. Scroll to CALCULATE and press ENTER