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Women in U.S. who have given birth μ > 25.4 POPULATION “Random Variable” X = Age at first birth mean μ = 25.4 H0:H0: “Null Hypothesis” μ < 25.4 That is, X ~ N(25.4, 1.5). μ < 25.4 standard deviation σ = 1.5 μ > 25.4 Year 2010: Suppose we know that X follows a “normal distribution” (a.k.a. “bell curve”) in the population. Or, is the “alternative hypothesis” H A : μ ≠ 25.4 true? mean Statistical Inference and Hypothesis Testing {x 1, x 2, x 3, x 4, …, x 400 } Study Question: Has “age at first birth” of women in the U.S. changed over time? public education, awareness programs socioeconomic conditions, etc. FORMULA Does the sample statistic tend to support H 0, or refute H 0 in favor of H A ? i.e., either or ? (2-sided) Present: Is μ = 25.4 still true?

IF H 0 is true, then we would expect a random sample mean to lie between and , with 95% probability. 95% CONFIDENCE INTERVAL FOR µ In order to answer this question, we must account for the amount of variability of different values, from one random sample of n = 400 individuals to another. BASED ON OUR SAMPLE DATA, the true value of μ today is between and , with 95% “confidence” (…akin to “probability”). We will see three things: % ACCEPTANCE REGION FOR H 0 IF H 0 is true, then we would expect a random sample mean that is at least 0.2 away from 25.4 (as ours was), to occur with probability (= 0.383%)… VERY RARELY!,which is less t “P-VALUE” of our sample THEORY EXPERIMENT

IF H 0 is true, then we would expect a random sample mean that is at least 0.2 away from 25.4 (as ours was), to occur with probability (= 0.383%)… VERY RARELY!,which is less t IF H 0 is true, then we would expect a random sample mean to lie between and , with 95% probability. 95% CONFIDENCE INTERVAL FOR µ In order to answer this question, we must account for the amount of variability of different values, from one random sample of n = 400 individuals to another. BASED ON OUR SAMPLE DATA, the true value of μ today is between and , with 95% “confidence” (…akin to “probability”). We will see three things: % ACCEPTANCE REGION FOR H 0 “P-VALUE” of our sample

IF H 0 is true, then we would expect a random sample mean that is at least 0.2 away from 25.4 (as ours was), to occur with probability (= 0.383%)… VERY RARELY!,which is less t IF H 0 is true, then we would expect a random sample mean to lie between and , with 95% probability. 95% CONFIDENCE INTERVAL FOR µ In order to answer this question, we must account for the amount of variability of different values, from one random sample of n = 400 individuals to another. BASED ON OUR SAMPLE DATA, the true value of μ today is between and , with 95% “confidence” (…akin to “probability”). We will see three things: % ACCEPTANCE REGION FOR H 0 “P-VALUE” of our sample O u r d a t a v a l u e l i e s i n t h e 5 % R E J E C T I O N R E G I O N. SIGNIFICANCE LEVEL (α) < Less than.05

IF H 0 is true, then we would expect a random sample mean that is at least 0.2 away from 25.4 (as ours was), to occur with probability (= 0.383%)… VERY RARELY!,which is less t IF H 0 is true, then we would expect a random sample mean to lie between and , with 95% probability. 95% CONFIDENCE INTERVAL FOR µ BASED ON OUR SAMPLE DATA, the true value of μ today is between and , with 95% “confidence” (…akin to “probability”) % ACCEPTANCE REGION FOR H 0 “P-VALUE” of our sample In order to answer this question, we must account for the amount of variability of different values, from one random sample of n = 400 individuals to another. We will see three things: Our data value lies in the 5% REJECTION REGION. SIGNIFICANCE LEVEL (α) < Less than.05 FORMAL CONCLUSIONS:  The 95% confidence interval corresponding to our sample mean does not contain the “null value” of the population mean, μ =  The 95% acceptance region for the null hypothesis does not contain the value of our sample mean,.  The p-value of our sample,.00383, is less than the predetermined α =.05 significance level. Based on our sample data, we may reject the null hypothesis H 0 : μ = 25.4 in favor of the two-sided alternative hypothesis H A : μ ≠ 25.4, at the α =.05 significance level. INTERPRETATION: According to the results of this study, there exists a statistically significant difference between the mean ages at first birth in 2010 (25.4 years old) and today, at the 5% significance level. Moreover, the evidence from the sample data suggests that the population mean age today is older than in 2010, rather than younger, by about 0.2 years. FORMAL CONCLUSIONS:  The 95% confidence interval corresponding to our sample mean does not contain the “null value” of the population mean, μ =  The 95% acceptance region for the null hypothesis does not contain the value of our sample mean,.  The p-value of our sample,.00383, is less than the predetermined α =.05 significance level. Based on our sample data, we may reject the null hypothesis H 0 : μ = 25.4 in favor of the two-sided alternative hypothesis H A : μ ≠ 25.4, at the α =.05 significance level. INTERPRETATION: According to the results of this study, there exists a statistically significant difference between the mean ages at first birth in 2010 (25.4 years old) and today, at the 5% significance level. Moreover, the evidence from the sample data suggests that the population mean age today is older than in 2010, rather than younger, by about 0.2 years.

SUMMARY: Why are these methods so important? They help to distinguish whether or not differences between populations are statistically significant, i.e., genuine, beyond the effects of random chance. They help to distinguish whether or not differences between populations are statistically significant, i.e., genuine, beyond the effects of random chance. Computationally intensive techniques that were previously intractable are now easily obtainable with modern PCs, etc. Computationally intensive techniques that were previously intractable are now easily obtainable with modern PCs, etc. If your particular field of study involves the collection of quantitative data, then eventually you will either: If your particular field of study involves the collection of quantitative data, then eventually you will either: 1 - need to conduct a statistical analysis of your own, or 1 - need to conduct a statistical analysis of your own, or 2 - read another investigator’s methods, results, and conclusions in a book or professional research journal. 2 - read another investigator’s methods, results, and conclusions in a book or professional research journal. Moral: You can run, but you can’t hide …. Moral: You can run, but you can’t hide ….