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Confidence Intervals with Means

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

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Confidence Intervals with Means

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  1. Confidence Intervals with Means

  2. Rate your confidence0 - 100 • Name my age within 10 years? • within 5 years? • within 1 year? • Shooting a basketball at a wading pool, will make basket? • Shooting the ball at a large trash can, will make basket? • Shooting the ball at a carnival, will make basket?

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

  4. Guess the number • Teacher will have pre-entered a number into the memory of the calculator. • Then, using the random number generator from a normal distribution, a sample mean will be generated. • Can you determine the true number?

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

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

  7. Margin of error • Shows howaccuratewe believe our estimate is • Thesmallerthe margin of error, themore preciseour estimate of the true parameter • Formula:

  8. Confidence level • Is the success rate of themethodused to construct an interval that contains that true mean • Using this method, ____% of the time the intervals constructedwillcontainthe true population parameter

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

  10. .05 .025 .005 Critical value (z*) • Found from the confidence level • The upper z-score with probability p lying to its right under the standard normal curve Confidence level tail area z* .05 1.645 .025 1.96 .005 2.576 z*=1.645 z*=1.96 z*=2.576 90% 95% 99%

  11. Confidence interval for a population mean: Standard deviation of thestatistic Critical value estimate Margin of error

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

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

  14. Confidence Interval Applet • http://bcs.whfreeman.com/tps4e/#628644__666391__ • The purpose of this applet is to understand how the intervals move but the population mean doesn’t.

  15. 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 σ = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level? • Assumptions: • Have an SRS of blood measurements • Potassium level is normally distributed (given) • s known • We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

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

  17. 99% confidence interval? • Assumptions: • Have an SRS of blood measurements • Potassium level is normally distributed (given) • s known • We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

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

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

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

  21. Suppose that we have this random sample of SAT scores: • 1130 1260 1090 1310 1420 1190 • What is a 95% confidence interval for the true mean SAT score? (Assume s = 105) We are 95% confident that the true mean SAT score for CHS students is between 1115.1 and 1270.6.

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

  23. The heights of CHS male students is normally distributed with σ = 2.5 inches. How large a sample is necessary to be accurate within +/- .75 inches with a 95% confidence interval? n = 43

  24. Student t-Distribution In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of 114.9 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? No, don’t know σ Only sample s

  25. Gossett Story

  26. Statistics are variables – each sample s will cause the shape to change away from a normal distribution Parameters are constant – don’t expect the shape to change, just shift based on changes in ẋ • Can you use s instead of σ when calculating a z-score (so that you can find the +/- 3 σ )? • Not exactly. Look at the two equations. ẋ has a normal distribution

  27. Do t-score bingo.

  28. 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

  29. T-Curves Basic properties of t-Curves • Property 1: The total area under a t-curve equals 1. • Property 2: A t-curve extends indefinitely in both directions, approaching the horizontal axis asymptotically • Property 3: A t-curve is symmetric about 0.

  30. T-curves continued • Property 4: As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve

  31. How does t compare to normal? • Shorter & more spread out • More area under the tails • As n increases, t-distributions become more like astandard normal distribution

  32. Graph examples of t- curve vs normal curve

  33. z – Score and t - Score

  34. Confidence Interval Formula: Standard deviation of statistic Critical value estimate Margin of error

  35. Can also use invT on the calculator! For 90% confidence level, 5% is above and 5% is below Need upper t* value with 5% above – so 0.95 is p value invT(p,df) How to find Margin of error when σ is not available – find t* • Use Table B for t distributions • Look up confidence level at bottom & degrees of freedom (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

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

  37. For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group. • Assumptions: • Have an SRS of healthy, white males • Systolic blood pressure is normally distributed (given). • s is unknown • We are 95% confident that the true mean systolic blood pressure is between 111.22 and 118.58.

  38. Robust • An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated. • For adequately sized samples (n≥30) , t-procedures can be used with some skewness, as long as there are no outliers.

  39. Ex. 5 – 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. (70.883, 74.497)

  40. 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.

  41. Ex. 6 – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: 160 200 220 230 120 180 140 130 170 190 80 120 100 170 Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. Boxplot shows approx normal distr. (126.16, 189.56)

  42. Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than! 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.

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

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

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