Let x represent the dollar amount spent on supermarket impulsebuying in a 10-minute (unplanned) shopping interval. Based on acertain article, the mean of the x distribution is about $31 andthe estimated standard deviation is about $8. (a) Consider a randomsample of n = 70 customers, each of whom has 10 minutes ofunplanned shopping time in a supermarket. From the central limittheorem, what can you say about the probability distribution of x,the average amount spent by these customers due to impulse buying?What are the mean and standard deviation of the x distribution? Thesampling distribution of x is not normal. The sampling distributionof x is approximately normal with mean ?x = 31 and standard error?x = $0.11. The sampling distribution of x is approximately normalwith mean ?x = 31 and standard error ?x = $8. The samplingdistribution of x is approximately normal with mean ?x = 31 andstandard error ?x = $0.96. Is it necessary to make any assumptionabout the x distribution? Explain your answer. It is not necessaryto make any assumption about the x distribution because ? is large.It is necessary to assume that x has a large distribution. It isnecessary to assume that x has an approximately normaldistribution. It is not necessary to make any assumption about thex distribution because n is large. (b) What is the probability thatx is between $29 and $33? (Round your answer to four decimalplaces.) (c) Let us assume that x has a distribution that isapproximately normal. What is the probability that x is between $29and $33? (Round your answer to four decimal places.) (d) In part(b), we used x, the average amount spent, computed for 70customers. In part (c), we used x, the amount spent by only onecustomer. The answers to parts (b) and (c) are very different. Whywould this happen? The standard deviation is smaller for the xdistribution than it is for the x distribution. The x distributionis approximately normal while the x distribution is not normal. Thesample size is smaller for the x distribution than it is for the xdistribution. The mean is larger for the x distribution than it isfor the x distribution. The standard deviation is larger for the xdistribution than it is for the x distribution. In this example, xis a much more predictable or reliable statistic than x. Considerthat almost all marketing strategies and sales pitches are designedfor the average customer and not the individual customer. How doesthe central limit theorem tell us that the average customer is muchmore predictable than the individual customer? The central limittheorem tells us that small sample sizes have small standarddeviations on average. Thus, the average customer is morepredictable than the individual customer. The central limit theoremtells us that the standard deviation of the sample mean is muchsmaller than the population standard deviation. Thus, the averagecustomer is more predictable than the individual customer.