Workshop Statistics: Discovery with Data and Fathom

Topic 15: Normal Distributions

Activity 15-1: Placement Scores and Hospital Births

(a) Both distributions are roughly symmetrical, mound shaped, and have a single peak.
(b) The student should draw smooth mound shaped curves that approximate the shapes of the histograms.
(c) A is the tall curve.  B is the dashed curve.  C is the solid line curve.
(d) Changing the mean shifts the curve horizontally.  Increasing the standard deviation narrows the curve, making it steeper with a higher peak.  Decreasing the standard deviation makes the curve more gradual and spread out, with a lower peak.
(e) yes
(f) z = (8.5-10.221)/3.859 = -.45
(g) z = (23.5-25.060)/3.472 = -.45
(h) They are equal.
(i) The two values fall .45 of one standard deviation below their respective means.
(j) (1+1+5+7+12+13+16+15)/213=.329
(k) 115/365=.315;  These proportions are fairly close.
(l) .329
(m) .3264
(n) yes
 

Activity 15-2: Birth Weights

(a) The student should shade in the area under the curve starting from the left, up to a vertical line representing 2500.
(b) Answers will vary from student to student.
(c) z=(2500-3250)/550= -1.36
(d) .0869
(e)

(f) z=(4536-3250)/550 = 2.34, proportion above= .0096
(g) 1. finding the area below z=2.34 (.9904) and subtracting from 1 to find area above.
     2. finding the area below -2.34 (.0096) since the curve is symmetric
(h) Shade between 3000 and 4000 in the curve.
Z(4000)=(4000-3250)/550 = 1.36, so proportion below = .9131
Z(3000) =(3000-3250)/550 = -.45 so proportion below = .3264
To find the area between we subtract: .9131-.3264 = .5867
(i) low birth weight proportion: .0750;  3000-4000 proportion: .6578;  Both calculations were off by less than .1.
(j) Shade the last 2.5% of curve to the left. Working backwards, we look .025 up in Table II and see this corresponds to z=-1.96 (negative since below the mean). Since z=-1.96 = (X-3250)/550, we can solve for X = -1.96(550)+3250 = 2172 grams. Thus, to be in the lightest 2.5% a baby’s birthweight needs to be 1.96 standard deviations below the mean, so take the mean (3250) and subtract 1.96 standard deviations (550) to get 2172 grams.
(k)

(l) The z-score is approximately 1.28, so the weight is 3250+1.28*550 = 3954 grams.
 

Activity 15-3: Matching Samples to Density Curves

(a) (b) (c) 100