Answer to Question #237027 in Statistics and Probability for Nana

Question

The annual sales of dealers of a company follows the normal distribution with its mean as

K94. A random sample of 10 dealers of the company is taken from the normal population.

The variance of the annual sales of these 10 dealers is K81. Find the probability that the

mean annual sales of the sample is

i. Less than K98

ii. More than K98

2

(b) The personnel manager of a company feels that 60% of the employees will have enhanced

skill after attending the training programme if they are sponsored for an in-house training

programme. A sample of records of 49 employees of the company, who attended the

training programme on skill development, reveals that only 24 of them are having enhanced

skill after attending the training programme. Find the probability that the sample of

employees who attended the training programme have enhanced their skill.

Answer 1

1.

(a)

Let $X=$ the mean annual sales: $X\sim N(\mu, \sigma^2/n)$

Given $\mu=94, n=10, \sigma^2/n=81$

i.

$$P(X<98)=P(Z<\dfrac{98-\mu}{\sqrt{\sigma^2/n}})$$

$$=P(Z<\dfrac{98-94}{\sqrt{81}})\approx P(Z<0.4444)$$

$$\approx0.67164$$

The probability that the mean annual sales of the sample is less than K98 is $0.67164.$

ii.

$$P(X>98)=1-P(X\leq 98)=1-P(X<98)$$

$$\approx 1-0.67164=0.32836$$

The probability that the mean annual sales of the sample is more than K98 is $0.32836.$

2.

(b)

$$p=\dfrac{24}{49}\approx0.4898$$

The probability that the sample of employees who attended the training programme have enhanced their skill is $0.4898.$