Basic Practice of Statistics - 3rd Edition Introduction to Inference Chapter 14 Introduction to Inference BPS - 5th Ed. Chapter 14 Chapter 13

Statistical Inference Provides methods for drawing conclusions about a population from sample data Confidence Intervals What is the population mean? Tests of Significance Is the population mean larger than 66.5? BPS - 5th Ed. Chapter 14

Inference about a Mean Simple Conditions SRS from the population of interest Variable has a Normal distribution N(m, s) in the population Although the value of m is unknown, the value of the population standard deviation s is known BPS - 5th Ed. Chapter 14

Confidence Interval A level C confidence interval has two parts An interval calculated from the data, usually of the form: estimate ± margin of error The confidence level C, which is the probability that the interval will capture the true parameter value in repeated samples; that is, C is the success rate for the method. BPS - 5th Ed. Chapter 14

Confidence Interval Mean of a Normal Population Take an SRS of size n from a Normal population with unknown mean m and known standard deviation s. A level C confidence interval for m is: BPS - 5th Ed. Chapter 14

Confidence Interval Mean of a Normal Population BPS - 5th Ed. Chapter 14

Reasoning of Tests of Significance What would happen if we repeated the sample or experiment many times? How likely would it be to see the results we saw if the claim of the test were true? Do the data give evidence against the claim? BPS - 5th Ed. Chapter 14

Stating Hypotheses Null Hypothesis, H0 The statement being tested in a statistical test is called the null hypothesis. The test is designed to assess the strength of evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference”, or it is a statement of equality. When performing a hypothesis test, we assume that the null hypothesis is true until we have sufficient evidence against it. BPS - 5th Ed. Chapter 14

Stating Hypotheses Alternative Hypothesis, Ha The statement we are trying to find evidence for is called the alternative hypothesis. Usually the alternative hypothesis is a statement of “there is an effect” or “there is a difference”, or it is a statement of inequality. The alternative hypothesis should express the hopes or suspicions we bring to the data. It is cheating to first look at the data and then frame Ha to fit what the data show. BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas Diet colas use artificial sweeteners to avoid sugar. These sweeteners gradually lose their sweetness over time. Trained testers sip the cola and assign a “sweetness score” of 1 to 10. The cola is then retested after some time and the two scores are compared to determine the difference in sweetness after storage. Bigger differences indicate bigger loss of sweetness. BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas Suppose we know that for any cola, the sweetness loss scores vary from taster to taster according to a Normal distribution with standard deviation s = 1. The mean m for all tasters measures loss of sweetness. The sweetness losses for a new cola, as measured by 10 trained testers, yields an average sweetness loss of = 1.02. Do the data provide sufficient evidence that the new cola lost sweetness in storage? BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas If the claim that m = 0 is true (no loss of sweetness, on average), the sampling distribution of from 10 tasters is Normal with m = 0 and standard deviation The data yielded = 1.02, which is more than three standard deviations from m = 0. This is strong evidence that the new cola lost sweetness in storage. If the data yielded = 0.3, which is less than one standard deviations from m = 0, there would be no evidence that the new cola lost sweetness in storage. BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas BPS - 5th Ed. Chapter 14

The Hypotheses for Means Null: H0: m = m0 One sided alternatives Ha: m > m0 Ha: m < m0 Two sided alternative Ha: m ¹ m0 BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas The null hypothesis is no average sweetness loss occurs, while the alternative hypothesis (that which we want to show is likely to be true) is that an average sweetness loss does occur. H0: m = 0 Ha: m > 0 This is considered a one-sided test because we are interested only in determining if the cola lost sweetness (gaining sweetness is of no consequence in this study). BPS - 5th Ed. Chapter 14

Studying Job Satisfaction Case Study II Studying Job Satisfaction Does the job satisfaction of assembly workers differ when their work is machine-paced rather than self-paced? A matched pairs study was performed on a sample of workers, and each worker’s satisfaction was assessed after working in each setting. The response variable is the difference in satisfaction scores, self-paced minus machine-paced. BPS - 5th Ed. Chapter 14

Studying Job Satisfaction Case Study II Studying Job Satisfaction The null hypothesis is no average difference in scores in the population of assembly workers, while the alternative hypothesis (that which we want to show is likely to be true) is there is an average difference in scores in the population of assembly workers. H0: m = 0 Ha: m ≠ 0 This is considered a two-sided test because we are interested determining if a difference exists (the direction of the difference is not of interest in this study). BPS - 5th Ed. Chapter 14

Test Statistic Testing the Mean of a Normal Population Take an SRS of size n from a Normal population with unknown mean m and known standard deviation s. The test statistic for hypotheses about the mean (H0: m = m0) of a Normal distribution is the standardized version of : BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas If the null hypothesis of no average sweetness loss is true, the test statistic would be: Because the sample result is more than 3 standard deviations above the hypothesized mean 0, it gives strong evidence that the mean sweetness loss is not 0, but positive. BPS - 5th Ed. Chapter 14

P-value Assuming that the null hypothesis is true, the probability that the test statistic would take a value as extreme or more extreme than the value actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence the data provide against the null hypothesis. That is, a small P-value indicates a small likelihood of observing the sampled results if the null hypothesis were true. BPS - 5th Ed. Chapter 14

P-value for Testing Means Ha: m > m0 P-value is the probability of getting a value as large or larger than the observed test statistic (z) value. Ha: m < m0 P-value is the probability of getting a value as small or smaller than the observed test statistic (z) value. Ha: m ¹ m0 P-value is two times the probability of getting a value as large or larger than the absolute value of the observed test statistic (z) value. BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas For test statistic z = 3.23 and alternative hypothesis Ha: m > 0, the P-value would be: P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006 If H0 is true, there is only a 0.0006 (0.06%) chance that we would see results at least as extreme as those in the sample; thus, since we saw results that are unlikely if H0 is true, we therefore have evidence against H0 and in favor of Ha. BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas BPS - 5th Ed. Chapter 14

Studying Job Satisfaction Case Study II Studying Job Satisfaction Suppose job satisfaction scores follow a Normal distribution with standard deviation s = 60. Data from 18 workers gave a sample mean score of 17. If the null hypothesis of no average difference in job satisfaction is true, the test statistic would be: BPS - 5th Ed. Chapter 14

Studying Job Satisfaction Case Study II Studying Job Satisfaction For test statistic z = 1.20 and alternative hypothesis Ha: m ≠ 0, the P-value would be: P-value = P(Z < -1.20 or Z > 1.20) = 2 P(Z < -1.20) = 2 P(Z > 1.20) = (2)(0.1151) = 0.2302 If H0 is true, there is a 0.2302 (23.02%) chance that we would see results at least as extreme as those in the sample; thus, since we saw results that are likely if H0 is true, we therefore do not have good evidence against H0 and in favor of Ha. BPS - 5th Ed. Chapter 14

Studying Job Satisfaction Case Study II Studying Job Satisfaction BPS - 5th Ed. Chapter 14

Statistical Significance If the P-value is as small as or smaller than the significance level a (i.e., P-value ≤ a), then we say that the data give results that are statistically significant at level a. If we choose a = 0.05, we are requiring that the data give evidence against H0 so strong that it would occur no more than 5% of the time when H0 is true. If we choose a = 0.01, we are insisting on stronger evidence against H0, evidence so strong that it would occur only 1% of the time when H0 is true. BPS - 5th Ed. Chapter 14

Tests for a Population Mean The four steps in carrying out a significance test: State the null and alternative hypotheses. Calculate the test statistic. Find the P-value. State your conclusion in the context of the specific setting of the test. The procedure for Steps 2 and 3 is on the next page. BPS - 5th Ed. Chapter 14

BPS - 5th Ed. Chapter 14

Case Study I Sweetening Colas Chapter 14 Hypotheses: H0: m = 0 Ha: m > 0 Test Statistic: P-value: P-value = P(Z > 3.23) = 1 – 0.9994 = 0.0006 Conclusion: Since the P-value is smaller than a = 0.01, there is very strong evidence that the new cola loses sweetness on average during storage at room temperature. BPS - 5th Ed. Chapter 14

Studying Job Satisfaction Case Study II Studying Job Satisfaction Hypotheses: H0: m = 0 Ha: m ≠ 0 Test Statistic: P-value: P-value = 2P(Z > 1.20) = (2)(1 – 0.8849) = 0.2302 Conclusion: Since the P-value is larger than a = 0.10, there is not sufficient evidence that mean job satisfaction of assembly workers differs when their work is machine-paced rather than self-paced. BPS - 5th Ed. Chapter 14