Question

The success rate of corneal transplant surgery is 90%. The surgery is performed on eight (n = 8) patients. Work on Excel

1. Construct a binomial distribution table for the above scenario; include columns for X and P(x).
2. Graph the probability histogram for this binomial distribution.
3. Extend the table with columns for x × P(x), (x – μ)², and (x – μ)² × P(x). Use this extended table to calculate the mean, variance, and standard deviation for this binomial distribution.
4. Find the probability that the surgery is successful for exactly four (X = 4) patients. Is this an unusual event? (Remember that an unusual event is one whose probability is < .05) Why or why not?
5. Find the probability that the surgery is successful for fewer than five (X < 5) patients. Is this an unusual event? Why or why not?

Answer 1

Result:

The success rate of corneal transplant surgery is 90%. The surgery is performed on eight (n = 8) patients. Work on Excel

    Construct a binomial distribution table for the above scenario; include columns for X and P(x).

Binomial Probabilities
Data
Sample size	8
Probability of an event of interest	0.9
Statistics
Mean = np =	7.2
Variance = np(1 - p)	0.7200
Standard deviation	0.8485
Binomial Probabilities Table
	X	P(X)
	0	0.0000
	1	0.0000
	2	0.0000
	3	0.0004
	4	0.0046
	5	0.0331
	6	0.1488
	7	0.3826
	8	0.4305

    Graph the probability histogram for this binomial distribution.

  

Extend the table with columns for x × P(x), (x – μ)2, and (x – μ)2 × P(x). Use this extended table to calculate the mean, variance, and standard deviation for this binomial distribution.

   

X	P(X)	x*(p(x)	(x-mean)^2	(x-mean)^2 *p(x)
0	0.0000	0.0000	51.84	5.184E-07
1	0.0000	0.0000	38.44	2.76768E-05
2	0.0000	0.0000	27.04	0.000613267
3	0.0004	0.0012	17.64	0.007201354
4	0.0046	0.0184	10.24	0.047029248
5	0.0331	0.1653	4.84	0.16004641
6	0.1488	0.8928	1.44	0.214277011
7	0.3826	2.6785	0.04	0.015305501
8	0.4305	3.4437	0.64	0.275499014
	Total	7.2000	152.16	0.72

Standard deviation = sqrt( 0.72) =0.8485

Find the probability that the surgery is successful for exactly four (X = 4) patients. Is this an unusual event? (Remember that an unusual event is one whose probability is < .05) Why or why not?

    P( x=4) = 0.0046

This is an unusual event because this p value is < 0.05.

Find the probability that the surgery is successful for fewer than five (X < 5) patients. Is this an unusual event? Why or why not?

P( x <5) =0.0050

This is an unusual event because this p value is < 0.05.