Essential Statistics Chapter 141 Thinking about Inference

What We’ll Learn? u Review Z-procedures u Conditions to apply Z-procedures u Margin of error discussion u Determine sample size n u P-value discussion u Discussion of statistical significance tests Essential Statistics Chapter 142

Essential Statistics Chapter 143  If we know the standard deviation  of the population, a confidence interval for the mean  is:  To test a hypothesis H 0 :  =  0 we use the one-sample z statistic: u These are called z procedures because they both involve a one-sample z statistic and use the standard Normal distribution. z Procedures

Essential Statistics Chapter 144 Conditions for Inference in Practice u The data must be an SRS from the population -- The z procedures are not correct for samples other than SRS. u Outliers can distort the result. –The sample mean is strongly influenced by outliers. –Always explore your data before performing an analysis u The shape of the population distribution matters. –Skewness and outliers make the z procedures untrustworthy unless the sample is large. –In practice, the z procedures are reasonably accurate for any sample of at least moderate size from a fairly symmetric distribution  The population standard deviation  must be known. –Unfortunately  is rarely known, so z procedures are rarely useful. –Chapter 16 will introduce procedures for when  is unknown.

Essential Statistics Chapter 145 u The margin of error is: u The margin of error gets smaller, resulting in more accurate inference, –when n gets larger –when z* gets smaller (confidence level gets smaller) –when  gets smaller (less variation) How Confidence Intervals Behave

Essential Statistics Chapter 146 Case Study NAEP Quantitative Scores (Ch. 14) 90% Confidence Interval The 90% CI is narrower than the 95% CI. 95% Confidence Interval

Essential Statistics Chapter 147 Cautions About Confidence Intervals The margin of error does not cover all errors. u The margin of error in a confidence interval covers only random sampling errors. No other source of variation or bias in the sample data influence the sampling distribution. u Practical difficulties such as undercoverage and nonresponse are often more serious than random sampling error. The margin of error does not take such difficulties into account.

Essential Statistics Chapter 148 Planning Studies Choosing the Sample Size for a C.I. The confidence interval for the mean of a Normal population will have a specified margin of error m when the sample size is:

Essential Statistics Chapter 149 Case Study NAEP Quantitative Scores (Ch.14) Suppose that we want to estimate the population mean NAEP scores using a 90% confidence interval, and we are instructed to do so such that the margin of error does not exceed 3 points (recall that  = 60). What sample size will be required to enable us to create such an interval?

Essential Statistics Chapter 1410 Case Study NAEP Quantitative Scores Thus, we will need to sample at least men aged 21 to 25 years to ensure a margin of error not to exceed 3 points. Note that since we can’t sample a fraction of an individual and using 1082 men will yield a margin of error slightly more than 3 points, our sample size should be n = 1083 men. Always round up!

Essential Statistics Chapter 1311 The five steps in carrying out a significance test: 1. State the null and alternative hypotheses. 2. Set significant level ɑ 3. Calculate the Z test statistic. 4. Find the P-value. 5. State your conclusion Review Tests Procedure for Population Mean

Essential Statistics Chapter 1312 Review The Hypotheses for Means  Null: H 0 :  =  0 u One sided alternatives H a :   H a :   u Two sided alternative H a :  0

Essential Statistics Chapter 1413 About Significance Tests one-sided tests versus two-sided tests u The P-value for a one-sided test is one-half the P-value for the two-sided test of the same null hypothesis based on the same data. The evidence against H 0 is stronger when the alternative is one-sided; use one-sided tests if you know the direction of possible deviations from H 0, otherwise you must use a two-sided alternative. P-value depends on the alternative hypothesis.

Essential Statistics Chapter 1414 About Significance Tests Sample Size Affects Statistical Significance ◙ Sample size (n). Other things being equal, the greater the sample size, the greater the power of the test. ◙ The confidence of the result are likely to increase with a higher sample size. ◙ If you want to reject your null hypothesis, then make sure your sample size is at least equal to the sample size calculated by the formula:null hypothesis Statistical significance is not the same thing as practical significance.

Essential Statistics Chapter 1415 u Alternative Hypothesis: The percentage of high school students who used alcohol in 1993 is less than the percentage who used alcohol in u Null Hypothesis: There is no difference in the percentage of high school students who used alcohol in 1993 and Alcohol Use Case Study

Essential Statistics Chapter survey was based on 17,000 seniors, 15,500 10th graders, and 18,500 8th graders. Grade DiffP-value 8 th < th th Case Study Alcohol Use

Essential Statistics Chapter 1417 u The article suggests that the survey reveals “good news” since the differences are all negative. u The differences are statistically significant. – All P-values are less than  = u The 10 th and 12 th grade differences probably are not practically significant. – Each difference is less than 1% Case Study Alcohol Use