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The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE

Chapter 8: Estimating with Confidence. Section 8.3 Estimating a Population Mean. The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE. Chapter 8 Estimating with Confidence. 8.1 Confidence Intervals: The Basics 8.2 Estimating a Population Proportion

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The Practice of Statistics, 4 th edition – For AP* STARNES, YATES, MOORE

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  1. Chapter 8: Estimating with Confidence Section 8.3 Estimating a Population Mean The Practice of Statistics, 4th edition – For AP* STARNES, YATES, MOORE

  2. Chapter 8Estimating with Confidence • 8.1 Confidence Intervals: The Basics • 8.2 Estimating a Population Proportion • 8.3 Estimating a Population Mean

  3. Estimating a Population Mean • Average Age of Pennies Suppose you want to estimate the average age of pennies in circulation. You take a random sample of 40 pennies, and find a mean age of 19.7 and standard deviation of 10.383. Can you create a 95% confidence interval to estimate the age of all pennies in circulation?

  4. P What parameter are we trying to estimate? How would you define it in context? SRS & n<10% of population, but how do we check to see if the sample size is large enough? A We’re dealing with means instead of proportions, so we know the name will be different N Statistic ± (critical value)(standard deviation of statistic) I z* ? C

  5. Estimating a Population Mean • When is Unknown: The t Distributions When we don’t know σ, we can estimate it using the sample standard deviation sx. What happens when we standardize? This new statistic does not have a Normal distribution!

  6. Estimating a Population Mean • When is Unknown: The t Distributions When we standardize based on the sample standard deviation sx, our statistic has a new distribution called a t distribution. It has a different shape than the standard Normal curve: • It is symmetric with a single peak at 0, • However, it has much more area in the tails. However, there is a different t distribution for each sample size, specified by its degrees of freedom (df).

  7. Properties of Student’s t- distribution • Developed by William Gosset (pseudonym “Student”) • Continuous distribution, …like the Normal model • Unimodal, symmetrical, bell-shaped density curve, …like the Normal model • Area under the curve equals 1, … like the Normal model • Based on degrees of freedom…. df = n - 1

  8. How does t compare to normal? • Shorter & more spread out (caused from using s as an estimate for ) • As n increases, t-distributions become more like a normal distribution

  9. How to find t* Can also use invT on the calculator! invT(percentile,df) • Use Table 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

  10. Standard deviation of statistic One-sample t-interval: Critical value Statistic (point estimate) Margin of error

  11. Assumptions for t-inference

  12. Example: In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups of equal size, where each group 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.

  13. P • → mean systolic blood pressure of healthy, white males • RAT given • 27 < 10% of all healthy white males • Nearly Normal? Yes – given that systolic blood pressure is normally distributed • We are 95% confident that the true mean systolic blood pressure of healthy white males is between 111.22 and 118.58. For the Example: Find a 95% confidence interval for the mean systolic blood pressure of healthy white males. A N 1-sample t-interval .95 27-1 = 26 2.056 I C = df = t* = C

  14. Ex. – 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. Conclusion: We are 98% confident that the true mean calorie content per serving of vanilla yogurt is between 126.16 & 189.56 calories.

  15. Ex– 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. Conclusion: We are 95% confident that the true mean pulse rate of adults is between 70.883 & 74.497 beats per minute.

  16. 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. Support 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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