Bluman A.G. Elementary Statistics- A Step By Step Approach

Section 6–4 The Normal Approximation to the Binomial Distribution 343Example 6–16Reading While DrivingA magazine reported that 6% of American drivers read the newspaper while driving. If300 drivers are selected at random, find the probability that exactly 25 say they read thenewspaper while driving.Source: USA Snapshot, USA TODAY.SolutionHere, p 0.06, q 0.94, and n 300.Step 1 Check to see whether a normal approximation can be used.np (300)(0.06) 18 nq (300)(0.94) 282Since np 5 and nq 5, the normal distribution can be used.Step 2 Find the mean and standard deviation.m np (300)(0.06) 18s npq3000.060.94 16.92 4.11Step 3 Write the problem in probability notation: P(X 25).Step 4 Rewrite the problem by using the continuity correction factor. Seeapproximation number 1 in Table 6–2: P(25 0.5 X 25 0.5) P(24.5 X 25.5). Show the corresponding area under the normaldistribution curve. See Figure 6–37.Figure 6–37Area Under a NormalCurve and X Values forExample 6–16251824.5 25.5Step 5 Find the corresponding z values. Since 25 represents any value between 24.5and 25.5, find both z values.25.5 1824.5 18z 1 1.82 z4.112 1.584.11Step 6 The area to the left of z 1.82 is 0.9656, and the area to the left of z 1.58 is0.9429. The area between the two z values is 0.9656 0.9429 0.0227, or2.27%. Hence, the probability that exactly 25 people read the newspaperwhile driving is 2.27%.Example 6–17Widowed BowlersOf the members of a bowling league, 10% are widowed. If 200 bowling leaguemembers are selected at random, find the probability that 10 or more will be widowed.SolutionHere, p 0.10, q 0.90, and n 200.Step 1 Since np (200)(0.10) 20 and nq (200)(0.90) 180, the normalapproximation can be used.6–45