Intro to Confidence Intervals Introduction to Inference Friday, April 12, 2019 Chapter 14 Introduction to Inference April 12, 2019 HS 67

Statistical Inference Two forms of statistical inference: Confidence intervals Significance tests of hypotheses April 12, 2019

Population Mean μ We wish to infer population mean μ using sample mean “x-bar”. The following conditions prevail: Data acquired by Simple Random Sample Population distribution is Normal The value of σ is known The value of μ is NOT known April 12, 2019

Example Statement of problem: Young people have a better chance of good jobs and wages if they are good with numbers. We want to know the average NAEP math score of a population. Response variable ≡ NAEP math scores Range from 0 to 500 Have a Normal distribution Population standard deviation σ = 60 Population mean μ not known A simple random sample of n = 840 individuals derives sample mean (“x-bar”) = 272 We want to estimate population mean µ Reference: Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults. Journal of Human Resources, 27, 313-328. April 12, 2019

The Sampling Distribution of the Mean and its Standard Deviation Sample means would vary from sample to sample, forming a sampling distribution This distribution will be Normal with mean μ The standard deviation of the distribution of means will equal population standard deviation σ divided by the square root of the sample size: This relationship is known as the square root law April 12, 2019

Sampling distribution of the mean For our example, σ = 60 and n = 840; thus, the sampling distribution of the mean will be Normal with April 12, 2019

Margin of Error for 95% Confidence The 68-95-99.7 rule says 95% of x-bars will fall in the interval μ ± 2σxbar More accurately, 95% will fall in μ ± 1.96∙σxbar 1.96∙σxbar is the margin of error m for 95% confidence April 12, 2019

In repeated independent samples: We call these intervals Confidence Intervals (CIs) April 12, 2019

How Confidence Intervals Behave April 12, 2019

Other Levels of Confidence Confidence intervals can be calculated at various levels of confidence by altering critical value z* : Common levels of confidence Confidence level C 90% 95% 99% Critical value z* 1.645 1.960 2.576 April 12, 2019

CI for μ when σ known To estimate μ with confidence level C, use Use Table C to look up z* values for various levels of confidence Confidence level C 90% 95% 99% Critical value z* 1.645 1.960 2.576 April 12, 2019

In this example: x-bar = 272 and σxbar = 2.1 Conf Interval for μ In this example: x-bar = 272 and σxbar = 2.1 For 99% confidence; use z* = 2.576 For 90% confidence, use z* = 1.645 April 12, 2019

Tests of Significance Recall: two forms of statistical inference Confidence intervals Hypothesis tests of significance The objective of confidence intervals is to estimate a population parameter The objective of a test of significance is to test a statistical hypothesis April 12, 2019

Tests of Significance: Reasoning Like with confidence intervals, we ask what would happen if we repeated the sample or experiment many times We assume the same conditions as before: (SRS, Normal population, population standard deviation σ known) We recognize that sample means will vary from sample to sample The text explains the reasoning behind tests of significance on pp. 369 – 376 April 12, 2019 Basics of Significance Testing 14

Tests of Statistical Significance: Procedure A. The claim is stated as a null hypothesis H0 and alternative hypotheses Ha Calculate a test statistic The test statistic is converted to a probability statement called a P-value The P-value is interpreted in terms of the weight of evidence against H0 April 12, 2019 Basics of Significance Testing 15

Example Example: We want to test whether data in a sample provides reliable evidence that a population has gained weight. Based on prior research, we know that weight gain in the population is Normal with σ = 1 lb We select n = 10 individuals The sample mean weight gain x-bar = 1.02 lbs. Is this level of evidence strong enough to conclude a population weight gain? April 12, 2019

Intro to Confidence Intervals HS 67 Friday, April 12, 2019 Friday, April 12, 2019 The Null Hypothesis The null hypothesis H0 is a statement of “no difference” in the form: H0: μ = μ0 where μ0 represents the value of the population if the null hypothesis is true In our example, μ0 = 0 and the null hypothesis is H0: μ = 0 Does the sample mean of 1.02 based on n = 10 provide strong evidence that the null hypothesis is untrue? The first step in the procedure is to state the hypotheses null and alternative forms. The null hypothesis (abbreviate “H naught”) is a statement of no difference. The alternative hypothesis (“H sub a”) is a statement of difference. Seek evidence against the claim of H0 as a way of bolstering Ha. The next slide offers an illustrative example on setting up the hypotheses. April 12, 2019 9: Basics of Hypothesis Testing 17 The Basics of Significance Testing HS 67 17

The Alternative Hypothesis The alternative hypothesis Ha contradicts the null hypothesis The alternative hypothesis can be stated in one of two ways The one-sided alternative specifies the direction of the difference For the current example, Ha: μ > 0, indicating a positive weight change in the population The two-sided alternative does not specific the direction of the difference For the current example Ha: μ ≠ 0, indicating a “weight change in the population” April 12, 2019 Basics of Significance Testing 18

Sampling Distribution of the Mean This is the sampling distribution of the mean if the null hypothesis is true (i.e., μ = 0) is shown here Note that the standard deviation of the mean is 4/12/2019 April 12, 2019 Basics of Significance Testing 19

Test Statistic for H0: μ = μ0 When population σ is known April 12, 2019 4/12/2019 Basics of Significance Testing 20

Test Statistic, Example For data example, x-bar = 1.02, n = 10, and σ = 1 4/12/2019 April 12, 2019 Basics of Significance Testing 21

Basics of Significance Testing P-Value from z table Convert z statistics to a P-value: For Ha: μ > μ0 P-value = Pr(Z > zstat) = right-tail beyond zstat For Ha: μ < μ0 P-value = Pr(Z < zstat) = left tail beyond zstat For Ha: μ ¹ μ0 P-value = 2 × one-tailed P-value 4/12/2019 April 12, 2019 Basics of Significance Testing 22

Basics of Significance Testing P-value: Example For current example, zstat = 3.23 and the one-sided P-value = Pr(Z > 3.23) = 1 − .9994 = .0006 Two-sided P-value = 2 × one-sided P = 2 × .0006 = .0012 April 12, 2019 Basics of Significance Testing 23

P-value: Interpretation P-value (definition) ≡ the probability the sample mean would take a value as extreme or more extreme than observed test statistic when H0 is true P-value (interpretation) Smaller-and-smaller P-values → stronger-and-stronger evidence against H0 P-value (conventions) .10 < P < 1.0  evidence against H0 is not significant .05 < P ≤ .10  evidence against H0 is marginally signif. .01 < P ≤ .05  evidence against H0 is significant 0 < P ≤ .01  evidence against H0 is highly significant April 12, 2019 Basics of Significance Testing 24

Basics of Significance Testing P-value: Example Let us interpret the current two-sided P-value of .0012 This provides strong evidence against H0 Thus, we reject H0 and say there is highly significant evidence of a weight gain in the population April 12, 2019 Basics of Significance Testing 25

Basics of Significance Testing Significance Level α ≡ threshold for “significance” If we choose α = 0.05, we require evidence so strong that it would occur no more than 5% of the time when H0 is true Decision rule P-value ≤ α  evidence is significant P-value > α  evidence not significant For example, let α = 0.01. The two-sided P-value = 0.0012 is less than .01, so data are significant at the α = .01 level April 12, 2019 Basics of Significance Testing 26

Summary one sample z test for a population mean April 12, 2019 Basics of Significance Testing 27