Introduction to Inference

Statistics 111 - Lecture 11 Introduction to Inference Sampling Distributions, Confidence Intervals and Hypothesis Testing Stat 111 - Lecture 11 - Confidence Intervals

Administrative Notes • No homework due Monday • Homework 4 will be due next Wednesday June 24 Stat 111 - Lecture 11 - Confidence Intervals

Sampling Distribution of Sample Mean • Distribution of values taken by statistic in all possible samples of size n from the same population • Assume: observations are independent and sampled from a population with mean  and variance 2 Sample 1 of size n Sample 2 of size n Sample 3 of size n Sample 4 of size n Sample 5 of size n Sample 6 of size n Sample 7 of size n Sample 8 of size n . . . Distribution of these values? Population Parameters:  and 2 Stat 111 - Lecture 11 - Confidence Intervals

Sampling Distribution of Sample Mean • The center of the sampling distribution of the sample mean is the population mean: • Over all samples, the sample mean will, on average, be equal to the population mean (no guarantees for 1 sample!) • The spread of the sampling distribution of the sample mean is • As sample size increases, variance of the sample mean decreases! • Central Limit Theorem: if the sample size is large enough, then the sample mean has an approximately Normal distribution • This is true no matter what the shape of the distribution of the original data! Stat 111 - Lecture 11 - Confidence Intervals

Confidence Intervals ? Population Parameter:  • Sample mean is the best estimate of  • However, we realize that the sample mean is probably not exactly equal to population mean, and that we would get a different value of the sample mean in another sample • Solution is to use our sample mean as the center of an entire interval of likely values for our population mean  Sampling Inference Estimation Sample Statistic: Stat 111 - Lecture 11 - Confidence Intervals

Example: Chips Ahoy • Nabisco guarantees that each 18 oz bag of Chips Ahoy contains at least 1,000 chips. • In late 1990’s, company challenged students to confirm their claim • Air Force Academy administered the study by sampling 42 bags from 275 sent by company Stat 111 - Lecture 11 - Confidence Intervals

Example: Chips Ahoy • Dataset: number of chips per bag in 42 sampled bags • What can we can we say about the average number of chips  in the “population” of all bags produced? • Our sample mean is 1216 chips per bag, but that is probably not exactly equal to our population mean • We need to use our sample mean to calculate an interval of likely values for our population mean Stat 111 - Lecture 11 - Confidence Intervals

Sampling Dist. for Sample Mean • We assume that each observation is an independent sample from the population with unknown mean  • We also assume (for now) that the population variance 2 is known to be equal to s2 = (117.5)2 • From last class, we know that the sampling distribution of the sample mean is Normal: Stat 111 - Lecture 11 - Confidence Intervals

Confidence Interval for  • From the 68-95-99.7 rule, we know that 95% of sample means should be within 2 SDs (36.2 chips) of the population mean  • So 95% of the time (or in 95% of the samples), the population mean  should be contained in the interval • ( - 36.2, + 36.2) = (1225.3,1297.7) • We call this a 95% confidence interval for  Stat 111 - Lecture 11 - Confidence Intervals

General Confidence Intervals for  • The 95% is called the confidence level of the interval • More generally, we can make an interval with any confidence level we want. The general formula for a 100·C % confidence interval for population mean  is: • is called the critical value of the interval. It is the value from a standard normal table that gives you a tail probability of (1-C)/2. SD( ) Stat 111 - Lecture 11 - Confidence Intervals

Example of Critical Values • 95% interval means that C=0.95 so 1-C = 0.05 • So we need a value Z* that gives us a tail probability (area under curve) of (1-C)/2 = 0.025 • Looking at Standard Normal Table, we see that Z* = 1.96 • In previous example, we rounded Z* to 2 Stat 111 - Lecture 11 - Confidence Intervals

Interpretation of confidence intervals • If many different samples from same population were collected (each giving us a 95% confidence interval for ), then 95% of these intervals should contain the true population mean  • Note that the interval for any one sample may not contain  ! • All confidence intervals have the same form: Estimate ± Margin of Error • For population means, the estimate is the sample mean and the margin of error is Z*·SD( ) where critical value Z* is determined by our confidence level Stat 111 - Lecture 11 - Confidence Intervals

Margin of Error and Sample Size • Before we sample our Chips Ahoy cookies bags, we want to decide the minimum number of bags needed for a certain margin of error (saves on cookies!) • Confidence intervals for population mean  have a margin of error which means • If we want a confidence level of 95% (so Z*=1.96) and we want a margin of error less than 100 chips, then so we need to sample at least 6 bags of chips. Stat 111 - Lecture 11 - Confidence Intervals

Sampling Distribution for Proportion • We also want to calculate confidence intervals for a population proportion p. • From last class, we know the sampling distribution of the sample proportion is also Normal: Stat 111 - Lecture 11 - Confidence Intervals

Confidence Intervals for Proportions • Based on the sampling distribution, our confidence interval for the population proportion p is • However, this interval is not very useful, since it still depends on the unknown population proportion p. • We fix this by using our sample proportion in place of p, so our 100·C% confidence interval for p is: Stat 111 - Lecture 11 - Confidence Intervals

Example: Gallup Poll • Sample of 2000 people asked if they will vote for McCain. Had a sample proportion of 0.51 for “yes.” • What is a 95% confidence interval for the true population proportion p of McCain voters? • In this example, n = 2000, = 0.51 and we use Z*=1.96 so our 95% confidence interval for p is: Interval contains values on both sides of 0.5, so we are not confident that McCain will win the election! Stat 111 - Lecture 11 - Confidence Intervals

Note of Caution • All of these confidence intervals were calculated using the normal distribution, which is justified by the central limit theorem • However, by doing this, we are making the assumptions that the sample size is large and the population variance is known! • We will see later how to calculate confidence intervals when these assumptions are not true Stat 111 - Lecture 11 - Confidence Intervals

Hypothesis Testing • Now use our sampling distribution results for a different type of inference: • testing a specific hypothesis • In some problems, we are not interested in calculating a confidence interval, but rather we want to see whether our data confirm a specific hypothesis • This type of inference is sometimes called statistical decision making, but the more common term is hypothesis testing Stat 111 - Lecture 11 - Hyp. Tests

Example: Blackout Baby Boom • New York City experienced a major blackout on November 9, 1965 • many people were trapped for hours in the dark and on subways, in elevators, etc. • Nine months afterwards (August 10, 1966), the NY Times claimed that the number of births were way up • They attributed the increased births to the blackout, and this has since become urban legend! • Does the data actually support the claim of the NY Times? • Using data, we will test the hypothesis that the birth rate in August 1966 was different than the usual birth rate Stat 111 - Lecture 11 - Hyp. Tests

Number of Births in NYC, August 1966 First two weeks • We want to test this data against the usual birth rate in NYC, which is 430 births/day Stat 111 - Lecture 11 - Hyp. Tests

Steps for Hypothesis Testing • Formulate your hypotheses: • Need a Null Hypothesis and an Alternative Hypothesis • Calculate the test statistic: • Test statistic summarizes the difference between data and your null hypothesis • Find the p-value for the test statistic: • How probable is your data if the null hypothesis is true? Stat 111 - Lecture 11 - Hyp. Tests

Null and Alternative Hypotheses • Null Hypothesis (H0) is (usually) an assumption that there is no effect or no change in the population • Alternative hypothesis (Ha) states that there is a real difference or real change in the population • If the null hypothesis is true, there should be little discrepancy between the observed data and the null hypothesis • If we find there is a large discrepancy, then we will reject the null hypothesis • Both hypotheses are expressed in terms of different values for population parameters Stat 111 - Lecture 11 - Hyp. Tests

Example: NYC blackout and birth rates • Let  be the mean birth rate in August 1966 • Null Hypothesis: • Blackout has no effect on birth rate, so August 1966 should be the same as any other month • H0:  = 430 (usual birth rate) • Alternative Hypothesis: • Blackout did have an effect on the birth rate • Ha: 430 • This is a two-sided alternative, which means that we are considering a change in either direction • We could instead use a one-sided alternative that only considers changes in one direction • Eg. only alternative is an increase in birth rate Ha: >430 Stat 111 - Lecture 11 - Hyp. Tests

Test Statistic • Now that we have a null hypothesis, we can calculate a test statistic • The test statistic measures the difference between the observed data and the null hypothesis • Specifically, the test statistic answers the question: “How many standard deviations is our observed sample value from the hypothesized value?” • For our birth rate dataset, the observed sample mean is 433.6 and our hypothesized mean is 430 • To calculate the test statistic, we need the standard deviation of our sample mean Stat 111 - Lecture 11 - Hyp. Tests

Sampling Distribution of Sample Mean • The center of the sampling distribution of the sample mean is the population mean: • The spread of the sampling distribution of the sample mean is • Central Limit Theorem: if the sample size is large enough, then the sample mean has an approximately Normal distribution Stat 111 - Lecture 11 - Hyp. Tests

Test Statistic for Sample Mean • Sample mean has a standard deviation of so our test statistic T is: • T is the number of standard deviations between our sample mean and the hypothesized mean • 0 is the notation we use for our hypothesized mean • To calculate our test statistic T, we need to know the population standard deviation  • For now we will make the assumption that  is the same as our sample standard deviation s • Later, we will correct this assumption! Stat 111 - Lecture 11 - Hyp. Tests

Test Statistic for Birth Rate Example • For our NYC births/day example, we have a sample mean of 433.6, a hypothesized mean of 430 and a sample standard deviation of 39.4 • Our test statistic is: • So, our sample mean is 0.342 standard deviations different from what it should be if there was no blackout effect • Is this difference statistically significant? Stat 111 - Lecture 11 - Hyp. Tests

Probability values (p-values) • Assuming the null hypothesis is true, the p-value is the probability we get a value as far from the hypothesized value as our observed sample value • The smaller the p-value is, the more unrealistic our null hypothesis appears • For our NYC birth-rate example, T=0.342 • Assuming our population mean really is 430, what is the probability that we get a test statistic of 0.342 or greater? Stat 111 - Lecture 11 - Hyp. Tests

Calculating p-values • To calculate the p-value, we use the fact that the sample mean has a normal distribution • Under the null hypothesis, the sample mean has a normal distribution with mean 0 and standard deviation so the test statistic: has a standard normal distribution! Stat 111 - Lecture 11 - Hyp. Tests

p-value for NYC dataset prob = 0.367 prob = 0.367 • If our alternative hypothesis was one-sided (Ha: >430), then our p-value would be 0.367 • Since are alternative hypothesis was two-sided our p-value is the sum of both tail probabilities • p-value = 0.367 + 0.367 = 0.734 T = -0.342 T = 0.342 Stat 111 - Lecture 11 - Hyp. Tests

Statistical Significance • If the p-value is smaller than , we say the data are statistically significant at level  • The most common -level to use is  = 0.05 • Next class, we will see that this relates to 95% confidence intervals! • The -level is used as a threshold for rejecting the null hypothesis • If the p-value < , we reject the null hypothesis that there is no change or difference Stat 111 - Lecture 11 - Hyp. Tests

Conclusions for NYC birth-rate data • The p-value = 0.734 for the NYC birth-rate data, so we can clearly not reject the null hypothesis at -level of 0.05 • Another way of saying this is that the difference between null hypothesis and our data is not statistically significant • So, we conclude that the data do not support the idea that there was a different birth rate than usual for the first two weeks of August, 1966. No blackout baby boom effect! Stat 111 - Lecture 11 - Hyp. Tests

Next Class - Lecture 12 • More Hypothesis Testing! • Moore, McCabe and Craig: Section 6.2-6.3 Stat 111 - Lecture 11 - Hyp. Tests

Introduction to Inference