Confidence Intervals with Means Chapter 9

What is the purpose of a confidence interval? To estimate an unknown population parameter

One-Sample z Confidence Interval for  2. The sample size n is large (generally n  30), and 3. , the population standard deviation, is known then the general formula for a confidence interval for a population mean  is given by

Formula: statistic Critical value Standard deviation of parameter Margin of error

Find a 90% confidence interval estimate for the true mean fills of catsup from this machine. Example A certain filling machine has a true population standard deviation  = ounces when used to fill catsup bottles. A random sample of 36 “6 ounce” bottles of catsup was selected from the output from this machine and the sample mean was ounces.

The z critical value is % Confidence Interval (5.955, 6.081) Conclusion: We are 90% confident that the true mean fills of catsup from the machine is between oz. and oz.

In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups. The participants either take calcium or a placebo. The paper reports a mean seated systolic blood pressure of with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed. Can you find a z-interval for this problem? Why or why not? We only know sample statistics! We do not know population standard deviation!

William S. Gossett Quality control engineer for Guiness Brewery in Dublin, Ireland Checked the stout’s quality by performing hypothesis tests Figured out a new family of models Student’s t distributions

Student’s t- distribution Developed by William Gosset Continuous distribution Unimodal, symmetrical, bell-shaped density curve Above the horizontal axis Area under the curve equals 1 Based on degrees of freedom df = n - 1

t Distributions

How does the t-distributions compare to the standard normal distribution? Bell-shaped and centered at 0 Shorter & more spread out More area under the tails As n increases, t-distributions become more like a standard normal distribution

Formula: statistic Critical value Standard deviation of statistic Margin of error Standard error – when you substitute s for .

Since each t distribution would require a table similar to the standard normal table, we usually only create a table of critical values for the t distributions. Appendix Table 3 in BOB t Distributions

How to find t* Use Table B for t distributions Look up confidence level at bottom & df on the sides df = n – 1 Find these t* 90% confidence when n = 5 95% confidence when n = 15 t* =2.132 t* =2.145 Can also use invT on the calculator! Need upper t* value with 5% is above – so 95% is below invT(p,df)

Let’s do some comparing: Finding probabilities Student’s t models:Normal model: Normalcdf(1.645, ∞) tcdf(1.645, ∞, 4) tcdf(1.645, ∞, 9) tcdf(1.645, ∞, 14) tcdf(1.645, ∞, 29) tcdf(1.645, ∞, 99) Student’s t models resemble normal models as sample size gets bigger…

Finding critical values: Student’s t models: Normal model: 95% invT(.975, 4) invT(.975, 9) invT(.975, 14) invT(.975, 29) invT(.975, 99) T critical values approach z critical values as sample size increases… z critical values: t critical values: invNorm(.975)

Steps for doing a confidence interval: 1)Identify by name or formula One-sample confidence interval for means 2)Assumptions 3)Calculate the interval 4)Write a statement about the interval in the context of the problem.

Assumptions for t-inference Have an SRS from population (or randomly assigned treatments)  unknown Normal (or approx. normal) distribution –Given –Large sample size –Check graph of data Use only one of these methods to check normality

Statement: (memorize!!) We are ________% confident that the true mean context is between ______ and ______.

Ex. 1) Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group. Assumptions: Have randomly assigned males to treatment Systolic blood pressure is normally distributed (given).  is unknown We are 95% confident that the true mean systolic blood pressure is between and

Ex. 2) A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults.

One-sample confidence interval for means Assumptions: random sample of adults Pulse rate is normally distributed (given).  is unknown We are 95% confident that the true mean pulse rate of adults is between &

Ex 2 continued) Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain. The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Ex. 3) Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. We are 98% confident that the true mean calorie content per serving of vanilla yogurt is between calories & calories.

Ex 3 continued) A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate? Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories. NOT EQUAL Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than!

Robust An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the normality assumption is violated. t-procedures can be used with some skewness, as long as there are no outliers. Larger n can have more skewness. Since there is more area in the tails in t- distributions, then, if a distribution has some skewness, the tail area is not greatly affected. CI & p-values deal with area in the tails – is the area changed greatly when there is skewness

Find a sample size: If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: Always round up to the nearest person!

Ex 4) The heights of SHS male students is normally distributed with  = 2.5 inches. How large a sample is necessary to be accurate within +.75 inches with a 95% confidence interval? n = 43

Some Cautions: The data MUST be a SRS from the population (or randomly assigned treatment) The formula is not correct for more complex sampling designs, i.e., stratified, etc. No way to correct for bias in data

Cautions continued: Outliers can have a large effect on confidence interval Must know  to do a z-interval – which is unrealistic in practice

Confidence Interval Example Ten randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are Find a 90% confidence interval estimate for the true mean number of hours of television watched per week by shut-ins.

t-critical value of by looking on the t table at 90% confidence with df = 9 or invt(.95, 9). Calculating the sample mean and standard deviation we have n = 10, = 85, s = Can we meet the assumptions for a t – confidence interval? (79.295, ) We are 90% confident that the true mean number of hours of television watched per week is between hours and hours.