Question

PLEASE USE R FOR THE FOLLOWING! I NEED THE CALCULATIONS FROM R TO THE NEAREST 0.0001 FOR PROBABILITIES.

2. A company producing fine crystal knows that 19% of its goblets have cosmetic flaws and
must be classified as “seconds.” Consider a random sample of 620 goblets.

In a – c use the Binomial distribution to compute the probability that

a. exactly 120 are flawed (use the d function)
b. at most 130 are flawed (use the p function – as this sums probabilities)
c. more than 100 are flawed.
d. Give values for the mean and standard deviation of the number of flawed goblets in such
samples of size 620. (Both to at least the nearest 0.01.)
e. What are the values of np (see part d) and nq (q = 1 – p)? Are both at least 10? If so, the
Normal distribution will produce good approximation to Binomial probabilities.
The answer to the second part of part above e is “Yes – both are at least 10.” So: Using the
Normal distribution obtain, an approximation to each of the following.
f. The probability in part a.
g. The probability in part b.
[The approximations in f – h will be quite good.]
h. The probability in part c.

Answer 1

a) Using d command we have

dbinom(120,620,0.19)

The required probability as

 0.0395

b) Using the command

pbinom(130,620,0.19)

we get the required probability as approximately

0.9021

c) Use command 1-pbinom(100, 620,0.19)

the required probability is approximately 0.9637

d) We know mean is given by

And standard deviation