Answer to Question #220515 in Statistics and Probability for lll

Question

For a large population of normally distributed account balances, the mean balance is µ  = 150.00 with standard deviation = $35.00. 

(i) What is the probability that one randomly sampled account has a balance that ​exceeds $160.00?  

(ii) What is the probability that the mean for a random sample of n = 40 accounts will exceed $160.00? 

Answer 1

(i) $X\sim N(150, 35^2)$

$$P(X>160)=1-P(X\leq160)$$

$$=1-P(X\leq \dfrac{160-150}{35})\approx0.387548$$

(ii) $\bar{X}\sim N(150, 35^2/40)$

$$P(\bar{X}>160)=1-P(\bar{X}\leq160)$$

$$=1-P(X\leq \dfrac{160-150}{35/\sqrt{40}})\approx0.035380$$