Stat 217 – Day 17 Topic 14: Sampling Distributions with quantitative data

Last Time – Confidence Interval for  Goal: Want to say something about population based on random sample, categorical variable When the Central Limit Theorem applies (simple random sample, large n relative to  ), the sample proportion should fall within 2 standard errors of the population proportion I’m 95% confident that the population proportion that would choose the right front tire is between.349 and.583.

What do I mean by that? Note: Technically I cannot say “there is a 95% chance that the population proportion is between.349 and.583”  What’s “random”? What’s changing? What’s being repeated?  Is why “invented” the word confidence… This method should work roughly 95% of the time in the long run (“95% confidence” indicates the reliability of the method) In what long run?

More formally Confidence interval for  : Wider with larger confidence levels Narrower with larger sample size Margin-of-error Determined by confidence level from Table II (p. 316) “standard error” Sample proportion = “half-width” of interval = length/2

Thought Question We talked before about whether a body temperature of would be surprising  Normal distribution  How much variation expect in body temperatures The standard of 98.6 has been called into question. A sample of 130 healthy adults had a mean body temperature of degrees. Does this convince you that 98.6 is not the mean body temperature of healthy adults?

Questioning 98.6 Is it a representative sample? What is the variation in body temperatures?  Sample standard deviation.733 degrees (s) What is the variation in sample means from random samples?  How far do we expect the sample mean to fall from the population mean just by chance?

Let’s do it all again Can we predict the behavior of sample means? Can we carry out a test of significance about the population mean? Can we construct a confidence interval for a population mean? Begin with a situation where have access to population…

Activity 14-1 (p. 275) Ages of Pennies  Skewed to the right with mean 12.26, SD = 9.61

n = 5 Sampling Distribution of sample mean  Same mean but very different shape and much less spread

n = 25 Sampling Distribution of sample mean  Same mean, even more symmetric (normal) and even smaller standard deviation

n = 50 Sampling distribution of sample mean  Same mean, even more normal, even smaller standard deviation (though not cut in half)

Something special about penny ages? Population Sampling distribution (n = 5, 25)

Central Limit Theorem for Sample Mean (p. 282) No matter what the population shape is, if the sample size (n) is large, the sampling distribution of sample means will be (approximately) normal with mean equal to the population mean and standard deviation equal to the formula  /  So same center but spread decreases from  as increase the number of observations in a sample  Will be exactly normal if population itself follows a normal distribution more skewed the population, larger n needs to be

Body Temperatures SD =.733/  130 =.064

To Turn in with Partner 1. Answer (g) and (h) 2. What time did you go to sleep last night? What time did you wake up? How many hours (to nearest quarter hour) did you get? For Thursday  Lab 5  Lab 6 pre-lab For “Monday”  Activity 14-4 (self-check)