Hi
Note: We can use the normal distribution as a close approximation to the
binomial distribution whenever np ≥ 5 and nq ≥ 5.
can use the normal distribution to approximate the binomial distribution
Yes, 25.5 appropriate (did say at most, that is including 25) See generic examples below
m = .68*50 = 34 Sd = 3.2985 (
z = (25.5-34)/3.2985= -2.58
P(z < -2.58) = .005
Below: z = 0, z = ± 1, z= ±2 , z= ±3 are plotted.
Note: z = 0 (x value the mean) 50% of the area under the curve is to the left and %50 to the right
Examples Using the normal distribution to approximate binomial distribution probabilities with n = 50 and p = .4
a) exactly 18
Left endpoint for 18 is x=17.5
z-score for left endpoint 17.5 is
(1) if your z-table reads from the middle, look up 0.72, get .2642
(2) if your z-table reads from the left, look up -0.72, get .2358
Right endpoint for 18 is x=18.5
z-score for right endpoint 18.5 is
(1) if your z-table reads from the middle, look up 0.43, get .1664
(2) if your z-table reads from the left, look up -0.43, get .3336
(1) if your z-table reads from the middle, subtract .2642-.1664 = .0978
(2) if your z-table reads from the left, subtract .3336-.2358 = .0978
Answer: .0978
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b) anywhere from 21-25 including end points
Left endpoint for 21 is x=20.5
z-score for left endpoint 20.5 is
(1) if your z-table reads from the middle, look up 0.14, get .0557
(2) if your z-table reads from the left, look up -0.72, get .5557
Right endpoint for 25 is x=25.5
z-score for right endpoint 25.5 is
(1) if your z-table reads from the middle, look up 1.59, get .4441
(2) if your z-table reads from the left, look up 1.59, get .9441
(1) if your z-table reads from the middle, subtract .4441-.0557 = .3884
(2) if your z-table reads from the left, subtract .9441-.5557 = .3884
Answer: .3884
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c) more than 23
That means the smallest possible value is 24
Left endpoint for 24 is x=23.5
z-score for left endpoint 23.5 is
(1) if your z-table reads from the middle, look up 1.01, get .3438
(2) if your z-table reads from the left, look up 1.01, get .8438
(1) if your z-table reads from the middle, subtract .5-.3438 = .1562
(2) if your z-table reads from the left, subtract 1-.8438 = .1562
z-score for right endpoint 25.5 is
Answer: .1562
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[Be careful here for part (d) coming up and other "more than" or "at least"
problems, if your table reads from the middle, then if the z-score is negative
you must ADD .5 not subtract.
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d) at least 17
That means the smallest possible value is 17
Left endpoint for 17 is x=16.5
z-score for left endpoint 16.5 is
(1) if your z-table reads from the middle, look up 1.01, get .3438
(2) if your z-table reads from the left, look up -1.01, get .1562
(1) if your z-table reads from the middle, add .5+.3438 = .8438,
because the z-score is negative.
(2) if your z-table reads from the left, subtract 1-.1562 = .8438
Answer: .8438