statistics

Question

(a) You are planning to make sales calls to 8 firms tomorrow. As a rough approximation, you figure out that each call has a 20% chance of resulting in a sale and that firms make their buying decisions without consulting each other. 1-find the probability of having a really terrible day with no sales at all. 2-find the probability of having no more than 2 sales. (b) The amount of time devoted to studying each week by students who receive a grade A in the course is normally distributed random variable with mean of 7.5 hours and standard deviation of 2.1 hours. 1-what proportion of these students study more than 10 hours 2-find the probability that such a student spends between 7 and 9 hours studying 3-what proportion of these students spend less than 3 hours studying 4-what is the amount of time below which 5% of all students spend studying (c) On a typical day your clothing store takes care of 2.7 special customers on average. These customers are taken directly to a special room in the back of the store, are assigned a fulltime serving assistant, and are given tea and scones, have clothes bought to them. 1-find the probability that no customers arrive on any given day 2-find the probability that at least 4 customers arrive on any given day

Answer 1

(a)(1) The probability of a sale is 0.20 so the probability of no sale is 0.80. The probability of no sales with 8 firms is 0.80^8=16.78% approx.

(2) We can use the binomial distribution with coefficients: 1 8 28 56 70 56 28 8 1. For no more than 2 sales we need the first 3 coefficients. (p+(p-1))^8=p^8+8p^7(1-p)+28p^6(1-p)^2+... where p=0.80. The first term is no sales at all, the second term just one sale, and the third term two sales. The total probability is 79.69%.

(b)(1) Z score for X=10 is (10-7.5)/2.1=1.19 (approx) corresponding to 88.30%. So the percentage of students studying more than 10hrs per week is 11.70%. (2) If X=9, Z=0.7143, corresponding to 76.25%; and  if X=7, Z=-0.2380, corresponding to 1-0.5940=40.60%. So the proportion of students spending 7-9 hours is 76.25-40.60=35.65% approx. (3) If X=3 then Z=-2.143, corresponding to 1-0.9839=0.0161 or about 1.61% of students spend less than 3 hrs a week studying. (4) In the distribution table Z=1.645 corresponding to 95%. Therefore Z=-1.645 corresponds to 5% and -1.645=(X-7.5)/2.1, X=7.5-3.4545=4.04 or about 4 hours. So 5% of students spend less than 4 hours studying per week.

(c) We don't know how many customers the clothing stores sees in one day. Should 2.7 be 2.7%?