Statistics The Normal Approximation to the Binomial Distribution
Example. Over the years, it has been observed that of all the lawyers who take the state bar exam, only 57% pass.
a. Suppose that this year 850 lawyers are going to take the Tennessee bar exam, how many are expected to pass? E = µ = np = 850 × .57 = 484.5 b. How many are expected to fail? 850 × .43 = 365.5
c. Can we use a normal distribution to approximate this binomial distribution? The tests: np ≥ 5 (YES, 484.5); nq ≥ 5 (YES, 365.5)
d. What is mean, variance and standard deviation of this distribution? µ = np = 484.5 σ2 = npq = 308.335 σ = 14.4338
e. What is the probability that 500 or more pass? Find this using a normal distribution and compare it to the binomial method discussed earlier. P(x ≼ 500) = P(x > 499.5) = P(z > 1.0392) = 0.1493 The second part is called correcting for continuity.
f. What is the probability that 500 or fewer pass? P(x â&#x2030;¤ 500) = P(x < 500.5) = P(z < 1.1085) = 0.8665
g. What is the probability that between 485 and 525, inclusive, pass? P(485 â&#x2030;¤ x â&#x2030;¤ 525) = P(484.5 < x < 525.5) = P(0 < z < 2.8406) = 0.4977
h. What is the probability that exactly 490 will pass? P(x = 490) = P(489.5 < x < 490.5) = P(0.3464 < z < 0.4157) = 0.0257
Guidelines: 1. Verify that the binomial distribution applies. 2. Determine if you can use the normal distribution to approximate the binomial variable. 3. Find the mean and the standard deviation for the distribution. 4. Apply the appropriate continuity correction. Shade the corresponding area under the normal curve. 5. Find the corresponding z-scores. 6. Find the probability.
The correction is to either add or subtract 0.5 of a unit from each discrete x-value. This fills in the gaps to make it continuous. This is very similar to expanding of limits to form boundaries that we did with group frequency distributions.
Discrete x=6 x>6 xâ&#x2030;Ľ6 x<6 xâ&#x2030;¤6
Continuous 5.5 < x < 6.5 x > 6.5 x > 5.5 x < 5.5 x < 6.5
Example. An engineering professional body estimates that 75% of the students taking undergraduate engineering courses are in favor of studying of statistics as part of their studies. If this estimate is correct, what is the probability that more than 780 undergraduate engineers out of a random sample of 1000 will be in favor of studying statistics?