ISES fllowino nrohleme ascrme \( r=.05 \), 2-tailed, and identify the following. EDPSY 490 Win 2025 Homework \( 6: 1 \)-group power \& 2-group NHST (Ch. 6-7) ( 15 points) (handwrite legibly) Due 4 pm Feb 18 th Name: 1. (c) Imagine you planned to increase your sample size to \( n=64 \) (instead of 16). Compute the new standard error. Then re- sketch the null and alternative. This will again be a picture of two sampling distributions that overlap. Use the steps discussed in class: (i) draw the null, (ii) label the critical regions, (iii) draw the alkernative, and (iv) find the area of the alternative that crosses into the critical region of the null, and shade that area. USE COLOR PENCILS.

Below is one acceptable answer. (Your handwritten version should show a neat bell‐curve drawing—with two overlapping normal curves in two different colors, along with all labels.) In your write‐up you would do the following: ────────────────────────────── Step 1. Compute the new standard error Recall that the standard error (SE) for a sample mean is   SE = σ/√n. If you originally had n = 16 then   SE₁ = σ/√16 = σ/4. Now with n = 64 the new standard error is   SE₂ = σ/√64 = σ/8. Notice that SE₂ is exactly half of SE₁, so the sampling distribution with n = 64 is “tighter” (narrower). ────────────────────────────── Step 2. Sketch the two sampling distributions You will need to produce a sketch with two parts:  (a) a “Null distribution” (the sampling distribution of the mean under H₀) and  (b) an “Alternative distribution” (the sampling distribution assuming the true effect is the nonzero value). a. Draw the null distribution:  • Sketch a bell‐shaped curve centered at 0 (if H₀ sets μ = 0).  • In a two‐tailed test with α = .05, mark the critical regions in both tails.   – If using a z-approximation, the critical z‐values are approximately ±1.96.   – Convert these to the measurement scale by multiplying by the new SE:    Critical values = 0 ± 1.96 · (σ/8).  • Label these boundaries “critical value(s)” (or “rejection regions”). Shade the two outer tail regions. b. Draw the alternative distribution:  • Sketch a second bell‐shaped curve in a different color (say blue if your null is black) with the same standard error (σ/8) but centered at the true mean (μₐ) that represents the effect. (The value of μₐ is based on your hypothesized effect size; for example, if you expect an increase of δ then center at δ.)  • Make clear that this curve overlaps with the null but is shifted to the right (or left, depending on the expected direction). ────────────────────────────── Step 3. Determine the shaded area (“power” region) The “power” of the test is the probability (under the alternative) of obtaining a value in the rejection region (the critical regions defined under the null). • For example, if your alternative distribution is centered at μₐ > 0, then the right‐tail rejection region (beyond 0 + 1.96 · σ/8) is of primary interest. • Shade the area under the alternative distribution that lies beyond the critical value (or values, if the shift goes in both directions). This shaded area equals the probability of a Type II error being avoided (i.e., the power).   – In more precise words: the area, under the alternative curve, that “crosses into” the rejection region of the null is your test’s power.   – Label this area “power” and indicate its approximate magnitude if possible (using standard normal areas). ────────────────────────────── Step 4. Write a short concluding note You might add: “By increasing n from 16 to 64, the standard error decreases from σ/4 to σ/8, which makes the alternative distribution narrower (and thus more separated from the null). The critical values (0 ± 1.96 · σ/8) are now closer to the center, so more of the alternative distribution falls into the rejection regions. This illustrated area represents the power of the test under the alternative hypothesis.” ────────────────────────────── Final Comments Be sure to use color pencils as requested:  • One color (say black) for the null distribution,  • A second color (say blue) for the alternative distribution,  • And clearly shade the rejection area (power area) under the alternative. Your drawing should have these clearly labeled parts:  – The null curve with its center at 0,  – The critical values at ±1.96 · (σ/8),  – The alternative curve centered at μₐ,  – And the shading of the part of the alternative curve that lies beyond the critical value(s). This complete answer shows that you understand that increasing the sample size decreases the standard error (making the sampling distribution more precise) and increases the overlap that qualifies for significant values (effectively increasing test power). ────────────────────────────── (End of sample answer.) Remember: You do not have to plug in a numerical value for σ if it is not provided; writing the answer in terms of σ is acceptable. If your course materials give a specific value for σ or an effect size (mean difference), you can substitute that to produce numerical critical values and areas.