Question

Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company’s management. The company guarantees the refund of money or a replacement for any calculator that malfunctions within two years from the date of purchase. It is known from past data that despite all efforts, 5% of the calculators manufactured by this company malfunction within a 2-year period. The company recently mailed 500 such calculators to its customers.

(a) Find the probability that exactly 29 of the 500 calculators will be returned for refund or replacement within a 2-year period.

(b) What is the probability that more than 26 of the 500 calculators will be returned for refund or replacement within a 2-year period?

(c) What is the probability that 15 to 22 of the 500 calculators will be returned for refund or replacement within a 2-year period?

Answer 1

SOLUTION:

here for expeced number of refund =np=500*0.05=25

std deviation=(np(1-p))^1/2 =4.8734

applying normal approximation of binomial distirbution

a) therefore probability =P(X=29)=P(28.5<X<29.5)=P((28.5-25)/4.8374<Z<(29.5-25)/4.8734)

=P(0.7182<Z<0.9234)=0.8221-0.7637=0.0584

b) P(X>26)=1-P(X<=26) =1-P(Z<(26.5-25)/4.8374)=1-P(Z<0.3078)=1-0.6209 =0.3791

c) P(15<=X<=22) =P((14.5-25)/4.8374<Z<(22.5-25)/4.8734)=P(-2.1546<Z<-0.5130)=0.3040-0.0156=0.2884