Appendi / A3. Statistical Inference / Mean, One Sample-1 A3. Statistical Inference Population Mean μ of a Random Variable with known standard deviation σ, and random sample of size n 1 Before selecting a random sample, the eperimenter first decides on each of the following Null Hypothesis H : μ = μ (the conjectured value of the true mean) Alternative Hypothesis 2 H A : μ μ (that is, either μ < μ or μ > μ ) Type I error Significance Level α = P (Reject H true) =.5, usually; therefore, H Confidence Level 1 α = P (Accept H H true) =, usually and calculates each of these following: Standard Error σ / n, the standard deviation of ; this is then used to calculate Margin of Error zα/2 σ / n, where the critical value z α/2 is computed via its definition: Z N (,1), P ( z α/2 < Z < z α/2) = 1 α, i.e., by tail-area symmetry, P (Z < z α/2) = P (Z > zα/2) = α /2. Note: If α =.5, then z.25 = 1.96. Figure 1 Illustration of the sample mean in the rejection region; note that p < α. Sampling Distribution N (μ, σ / n ) p / 2 p / 2 margin of error Acceptance Region for H Rejection Region for H 1 If σ is unknown, but n 3, then estimate σ by the sample standard deviation s. If n < 3, then use the t-distribution instead of the standard normal Z-distribution. 2 The two-sided alternative is illustrated here. Some formulas may have to be modified slightly if a one-sided alternative is used instead. (Discussed later )
Appendi / A3. Statistical Inference / Mean, One Sample-2 After selecting a random sample, the eperimenter net calculates the statistic Sample Mean = point estimate of μ then calculates any or all of the following: (1 α) 1% Confidence Interval the interval centered at, such that P ( μ inside) = 1 α C.I. = ( margin of error, + margin of error) = interval estimate of μ Decision Rule At the (1 α) 1% confidence level If μ is contained in C.I., then accept H. If μ is not in C.I., then reject H in favor of H A. (1 α) 1% Acceptance Region the interval centered at μ, such that P ( inside) = 1 α A.R. = (μ margin of error, μ + margin of error) Decision Rule If is in the acceptance region, then accept H. If is not in the acceptance region (i.e., is in the rejection region), then reject H in favor of H A. SEE FIGURE 1! p-value a measure of how significantly our sample mean differs from the null hypothesis p = the probability of obtaining a random sample mean that is AS, or MORE, etreme than the value of actually obtained, assuming the null hypothesis H : μ = μ is true. = P (obtaining a sample mean on either side of μ, as far away as, or farther than, is ) = P Z μ σ / n + P μ Z σ / n Left-sided area + Right-sided area cut by both and its symmetrically reflected value through μ NOTE: By symmetry, can multiply the amount of area in one tail by 2. Decision Rule If p < α, then reject H in favor of H A. The difference between and μ is statistically significant. If p > α, then accept H. The difference between and μ is not statistically significant. SEE FIGURE 1!
Appendi / A3. Statistical Inference / Mean, One Sample-3 For a one-sided hypothesis test, the preceding formulas must be modified. The decision to reject H in favor of H A depends on the probability of a sample mean being either significantly larger, or significantly smaller, than the value μ (always following the direction of the alternative H A ), but not both, as in a two-sided test. Previous remarks about σ and s, as well as z and t, still apply. Hypotheses (Case 1) H : μ μ vs. H A : μ > μ, right-sided alternative Confidence Interval = ( z α (σ / n ), + ) Acceptance Region = (, μ + z α (σ / n ) ) p-value = P (obtaining a sample mean that is equal to, or larger than, ) = P Z μ σ /, right-sided area cut off by (darkened) n Figure 2 Illustration of the sample mean in the rejection region; note that p < α. Decision Rule If p < α, then is in rejection region for H. is significantly larger than μ. p If p > α, then is in acceptance region for H. is not significantly larger than μ. Acceptance Region Rejection Region Hypotheses (Case 2) H : μ μ vs. H A : μ < μ Confidence Interval = (, + z α (σ / n ) ) Acceptance Region = ( μ z α (σ / n ), + ) p-value = P (obtaining a sample mean that is equal to, or smaller than, ) = P Z μ σ /, left-sided area cut off by (darkened) n, left-sided alternative Figure 3 Illustration of the sample mean in the rejection region; note that p < α. p Decision Rule If p < α, then is in rejection region for H. is significantly smaller than μ. If p > α, then is in acceptance region for H. is not significantly smaller than μ. Rejection Region Acceptance Region
Appendi / A3. Statistical Inference / Mean, One Sample-4 Eamples Given: Assume that the random variable = IQ score is normally distributed in a certain study population, with standard deviation σ = 3., but with unknown mean μ. Conjecture a null hypothesis H: μ = 1 vs. the (two-sided) alternative hypothesis H A : μ 1. Figure 4 Normal distribution of = IQ score, under the conjectured null hypothesis H : μ = 1 N(1, 3) 3 1 Question: Do we accept or reject H at the 5% (i.e., α =.5) significance level, and how strong is our decision, relative to this 5%? Suppose statistical inference is to be based on random sample data of size n = 4 individuals. Procedure: Decision Rule will depend on calculation of the following quantities. First, Standard Normal Distribution N(, 1) Margin of Error = Critical Value Standard Error = z α / 2 σ / n.25.25 = 1.96 3 / 4 if α =.5 given 1.96 1.96 Z = 1.96 1.5 = 2.94 then, THEORY EPERIMENT compare hypothesized mean value (1) μ...with mean of random sample data
Acceptance Region for Appendi / A3. Statistical Inference / Mean, One Sample-5 : All values between 1 ± 2.94, i.e., (97.6, 12.94). Figure 5 Null Distribution Sampling distribution of, under the null hypothesis H : μ = 1. Compare with Figure 4 above. Why? N(1, 1.5).25.25 97.6 1 11 12.94 15 Sample # 1: Suppose it is found that = 15 (or 95). As shown in Figure 5, this value lies far inside the α =.5 rejection region for the null hypothesis H (i.e., true mean μ = 1). In particular, we can measure eactly how significantly our sample evidence differs from the null hypothesis, by calculating the probability (that is, the area under the curve) that a random sample mean will be as far away, or farther, from μ = 1 as = 15 is, on either side. Hence, this corresponds to the combined total area contained in the tails to the left and right of 95 and 15 respectively, and it is clear from Figure 5 that this value will be much smaller than the combined shaded area of.5 shown. This can be checked by a formal computation: p-value = P ( 95) + P ( 15) by definition = 2 P ( 95), by symmetry = 2 P (Z 3.33), since (95 1) / 1.5 = 3.33, to two places = 2.4, via tabulated entry for N(,1) tail areas =.8 <<.5, statistically significant difference As observed, our p-value of.8% is much smaller than the accepted 5% probability of committing a Type I error (i.e., the α =.5 significance level) initially specified. Therefore, as suggested above, this sample evidence indicates a strong rejection of the null hypothesis at the.5 level. As a final method of verifying this decision, we may also calculate the sample-based 95% Confidence Interval for μ : All values between 15 ± 2.94, i.e., (12.6, 17.94). μ = 1 12.6 = 15 17.94 By construction, this interval should contain the true value of the mean μ, with 95% confidence. Because μ = 1 is clearly outside the interval, this shows a reasonably strong rejection at α =.5.
Similarly, we can eperiment with means Appendi / A3. Statistical Inference / Mean, One Sample-6 that result from other random samples. Sample # 2: Suppose it is now found that = 13 (or likewise, 97). This sample mean is closer to the hypothesized mean μ = 1 than that of the previous sample, hence it is somewhat stronger evidence in support of H. However, Figure 5 illustrates that 13 is only very slightly larger than the rejection region endpoint 12.94, thus we technically have a borderline rejection of H at α =.5. In addition, we can see that the combined left and right tail areas will total only slightly less than the.5 significance level. Proceeding as above, p-value = P ( 97) + P ( 13) by definition = 2 P ( 97), by symmetry = 2 P (Z 2), since (97 1) / 1.5 = 2 = 2.228, via tabulated entry of N(,1) tail areas =.456.5, borderline significant difference Finally, additional insight may be gained via the 95% Confidence Interval for μ : All values between 13 ± 2.94, i.e., (1.6, 15.94). μ = 1 1.6 = 13 15.94 As before, this interval should contain the true value of the mean μ with 95% confidence, by definition. Because μ = 1 is just outside the interval, this shows a borderline rejection at α =.5. Sample # 3: Suppose now = 11 (or 99). The difference between this sample mean and the hypothesized mean μ = 1 is much less significant, hence this is quite strong evidence in support of H. Figure 5 illustrates that 11 is clearly in the acceptance region of H at α =.5. Furthermore, p-value = P ( 99) + P ( 11) by definition = 2 P ( 99), by symmetry = 2 P (Z.67), since (99 1) / 1.5 =.67, to two places = 2.2514, via tabulated entry of N(,1) tail areas =.528 >>.5, not statistically significant difference 95% Confidence Interval for μ : All values between 11 ± 2.94, i.e., (98.6, 13.94). As before, this interval should contain the true value of μ with 95% confidence, by definition. Because μ = 1 is clearly inside, this too indicates acceptance of H at the α =.5 level. Other Samples: As an eercise, show that if = 1.3, then p =.8414; if = 1.1, then p =.9442, etc. From these eamples, it is clear that the closer the random sample mean gets to the hypothesized value of the true mean μ, the stronger the empirical evidence is for that hypothesis, and the higher the p-value. (Of course, the maimum value of any probability is 1.)
Appendi / A3. Statistical Inference / Mean, One Sample-7 Net suppose that, as before, = IQ score is normally distributed, with σ = 3., and that statistical inference for μ is to be based on random samples of size n = 4, at the α =.5 significance level. But perhaps we now wish to test specifically for significantly higher than average IQ in our population, by seeing if we can reject the null hypothesis H : μ 1, in favor of the (right-sided) alternative hypothesis H A : μ > 1, via sample data. Proceeding as before (with the appropriate modifications), we have Standard Normal Distribution N(, 1) Margin of Error = Critical Value Standard Error = z α σ / n.5 if α =.5 = 1.645 1.5 1.645 Z = 2.4675 Figure 6 Null Distribution N(1, 1.5).5 1 12.4675 Acceptance Region for : All values below 1 + 2.4675, i.e., < 12.4675. Samples: As in the first eample, suppose that = 15, which is clearly in the rejection region. The corresponding p-value is P ( 15), i.e., the single right-tailed area only, or.4 eactly half the two-sided p-value calculated before. (Of course, this leads to an even stronger rejection of H at the α =.5 level than before.) Likewise, if, as in the second sample, = 13, the corresponding p-value =.228 <.5, a moderate rejection. The sample mean = 11 is in the acceptance region, with a right-sided p-value =.2514 >.5. Clearly, = 1 corresponds to p =.5 eactly; = 99 corresponds to p =.7486 >>.5, and as sample means continue to decrease to the left, the corresponding p-values continue to increase toward 1, as empirical evidence in support of the null hypothesis H : μ 1 continues to grow stronger.