Question

Report results to 5 significant figures.

Show all units of measurement as appropriate.

3. What (a,b) are the two parameters of any binomial distribution?  Using these parameters, what (c, d) are the expressions used for calculating the mean and standard deviation of a binomial distribution?  In order to be able to approximate a binomial distribution using a Normal distribution, (e, f) what are the two criteria that must be met?  In one of Gregor Mendel’s genetics experiments, he expected 75% of pea plants resulting from a particular cross to bear green pods.  (g) What was the expected proportion of plants bearing green pods?  (h) What was the expected proportion of plants bearing non-green pods?  If he had a total of 580 plants in this particular experiment, (i) what number of plants would he have expected to bear green pods?  In fact, 428 plants bore green pods.  First, (j, k) check the validity of using a Normal approximation to the binomial for the data from this experiment.  Then, assumingp= 0.75 and using a Normal approximation to the binomial, (l) calculate the probability of obtaining exactly 428 plants bearing green pods.  Since the result of 428 is fewer than 435, (m) calculate the probability that, out of 580 plants, 428 or fewer would bear green pods, again assuming that p= 0.75.  (Observe the need to use the continuity correction for parts (l) and (m)).   (n) Is the probability of 428 or fewer plants bearing green pods unusual, in a statistical sense?  (o) Which probability calculated above is more useful in determining whether Mendel’s claimed proportion of 75% is incorrect (state “l” or “m”).                               15        

Answer 1

R-commands and outputs:
> 0.75*580
[1] 435
> p=0.75
> q=1-p
> n=580
> pbinom(428,n,p)
[1] 0.2649558
> O=428
> Z=(O-n*p)/sqrt(n*p*q)
> Z
[1] -0.6712486
> pnorm(Z)
[1] 0.2510311

### Rough
> pbinom(440,n,p)
[1] 0.6990141
> pbinom(441,n,p)
[1] 0.7318279
> pbinom(442,n,p)
[1] 0.7627857