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Confidence Intervals Chapter 10

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Rate your confidence 0 - 100 0 Name my age within 10 years? 0 within 5 years? 0 within 1 year? 0 Shooting a basketball at a wading pool, will make basket? 0 Shooting the ball at a large trash can, will make basket? 0 Shooting the ball at a carnival, will make basket?

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What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval. Simulation

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Point Estimate single 0 Use a single statistic based on sample data to estimate a population parameter 0 Simplest approach variation 0 But not always very precise due to variation in the sampling distribution

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Confidence intervals 0 Are used to estimate the unknown population mean 0 Formula: estimate + margin of error

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Margin of error 0 Shows how accurate we believe our estimate is more precise 0 The smaller the margin of error, the more precise our estimate of the true parameter 0 Formula:

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Confidence level 0 Is the success rate of the method used to construct the interval 0 Using this method, ____% of the time the intervals constructed will contain the true population parameter

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Critical value (z*) 0 Found from the confidence level 0 The upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz*.051.645.0251.96.0052.576.05 z*=1.645.025 z*=1.96.005 z*=2.576 90% 95% 99%

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What does it mean to be 95% confident? 0 95% chance that is contained in the confidence interval 0 The probability that the interval contains is 95% 0 The method used to construct the interval will produce intervals that contain 95% of the time.

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Confidence interval for a population mean: estimate Critical value Standard deviation of the statistic Margin of error

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Steps for doing a confidence interval: 1) Assumptions – SRS from population Sampling distribution is normal (or approximately normal) 0 Given (normal) 0 Large sample size (approximately normal) 0 Graph data (approximately normal) is known 2) Calculate the interval 3) Write a statement about the interval in the context of the problem.

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Statement: (memorize!!) We are ________% confident that the true mean context lies within the interval ______ and ______.

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Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) known We are 90% confident that the true mean potassium level is between 3.01 and 3.39. A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level?

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Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) known We are 95% confident that the true mean potassium level is between 2.97 and 3.43. 95% confidence interval?

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99% confidence interval? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) known We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

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What happens to the interval as the confidence level increases? the interval gets wider as the confidence level increases

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Critical value (z*) 0 Found from the confidence level 0 The upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz*.051.645.0251.96.0052.576.05 z*=1.645.025 z*=1.96.005 z*=2.576 90% 95% 99%

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How can you make the margin of error smaller? 0 z* smaller (lower confidence level) 0 smaller (less variation in the population) 0 n larger (to cut the margin of error in half, n must be 4 times as big) Really cannot change!

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A random sample of 50 BGHS students was taken and their mean SAT score was 1250. (Assume = 105) What is a 95% confidence interval for the mean SAT scores of BGHS students? We are 95% confident that the true mean SAT score for BGHS students is between 1220.9 and 1279.1

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Find a sample size: 0 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!

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The heights of BGHS 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 Homework pg. 632-633 7- 11, 13, 15

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t- distribution 0 Developed by William Gosset 0 Continuous distribution 0 Unimodal, symmetrical, bell-shaped density curve 0 Above the horizontal axis 0 Area under the curve equals 1 0 Based on degrees of freedom

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Graph examples of t- curves vs normal curve

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

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How to find t* 0 Use Table B for t distributions 0 Look up confidence level at bottom & df on the sides 0 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)

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

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Assumptions for t-inference 0 Have an SRS from population 0 unknown 0 Normal distribution 0 Given 0 Large sample size 0 Check graph of data

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Robust 0 An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated. 0 t-procedures can be used with some skewness, as long as there are no outliers. 0 Larger n can have more skewness.

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0 Outliers are always a concern, but they are even more of a concern for confidence intervals using the t-distribution 0 Sample mean is not resistant; hence the sample mean is larger or smaller (drawn toward the outlier) (small numbers of n in t-distribution!) 0 Sample standard deviation is not resistant; hence the sample standard deviation is larger 0 Confidence intervals are much wider with an outlier included 0 Options: 0 Make sure data is not a typo (data entry error) 0 Increase sample size beyond 30 observations

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A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 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. We are 95% confident that the true mean pulse rates of adults is between 70.883 and 74.497 beat per minute.

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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.

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Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: 160200220230120180140 13017019080120100170 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 is between 126.16 and 189.56 calories.

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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!

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Some Cautions: 0 The data MUST be a SRS from the population 0 The formula is not correct for more complex sampling designs, i.e., stratified, etc. 0 No way to correct for bias in data

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Cautions continued: 0 Outliers can have a large effect on confidence interval 0 Must know to do a z-interval – which is unrealistic in practice

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0 Homework: 0 10.27, 28, 29 0 Pg.648-649

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