(5) Suppose x has a distribution with ? = 20and ? = 16.
(a) If a random sample of size n = 47 is drawn, find?x, ?xand P(20 ? x ? 22). (Round?x to two decimal places and theprobability to four decimal places.)
?x =
? x =
P(20 ? x ? 22)=
(b) If a random sample of size n = 61 is drawn, find?x, ?xand P(20 ? x ? 22). (Round?x to two decimal places and theprobability to four decimal places.)
?x =
? x =
P(20 ? x ? 22)=
c) Why should you expect the probability of part (b) to behigher than that of part (a)? (Hint: Consider the standarddeviations in parts (a) and (b).)
The standard deviation of part (b) is (Blank)? part (a)because of the ( Blank) ? Sample size. Therefore, the distributionabout ?x is (Blank) ?
(8) Let x be a random variable that represents thelevel of glucose in the blood (milligrams per deciliter of blood)after a 12 hour fast. Assume that for people under 50 years old,x has a distribution that is approximately normal, withmean ? = 54 and estimated standard deviation ? =11. A test result x < 40 is an indication of severeexcess insulin, and medication is usually prescribed.
(a) What is the probability that, on a single test, x< 40? (Round your answer to four decimal places.)
(b) Suppose a doctor uses the average x for two tests takenabout a week apart. What can we say about the probabilitydistribution of x? Hint: See Theorem 6.1.
What is the probability that x < 40? (Round your answer tofour decimal places.)
(c) Repeat part (b) for n = 3 tests taken a week apart.(Round your answer to four decimal places.)
(d) Repeat part (b) for n = 5 tests taken a week apart.(Round your answer to four decimal places.)
Explain what this might imply if you were a doctor or anurse.
(9) Let x represent the dollar amount spent onsupermarket impulse buying in a 10-minute (unplanned) shoppinginterval. Based on a certain article, the mean of the xdistribution is about $27 and the estimated standard deviation isabout $9.
(a) Consider a random sample of n = 100 customers, eachof whom has 10 minutes of unplanned shopping time in a supermarket.From the central limit theorem, what can you say about theprobability distribution of x, the average amount spent bythese customers due to impulse buying? What are the mean andstandard deviation of the x distribution?
Is it necessary to make any assumption about the xdistribution? Explain your answer.
(b) What is the probability that x is between $25 and$29? (Round your answer to four decimal places.)
(c) Let us assume that x has a distribution that isapproximately normal. What is the probability that x isbetween $25 and $29? (Round your answer to four decimalplaces.)
(d) In part (b), we used x, the average amountspent, computed for 100 customers. In part (c), we used x,the amount spent by only one customer. The answers toparts (b) and (c) are very different. Why would this happen?
(10) A European growth mutual fund specializes in stocks fromthe British Isles, continental Europe, and Scandinavia. The fundhas over 325 stocks. Let x be a random variable thatrepresents the monthly percentage return for this fund. Supposex has mean ? = 1.4% and standard deviation? = 1.1%.
(a) Let's consider the monthly return of the stocks in the fundto be a sample from the population of monthly returns of allEuropean stocks. Is it reasonable to assume that x (theaverage monthly return on the 325 stocks in the fund) has adistribution that is approximately normal? Explain.
(Blank) x is a mean of a sample ofn = 325 stocks. By the(Blank) the xdistribution( Blank) approximately normal?
(b) After 9 months, what is the probability that theaverage monthly percentage return x will be between 1% and2%? (Round your answer to four decimal places.)(c)After 18 months,what is the probability that the average monthlypercentage return x will be between 1% and 2%? (Round your answerto four decimal places.
(d) Compare your answers to parts (b) and (c). Did theprobability increase as n (number of months) increased?Why would this happen?
(e) If after 18 months the average monthly percentage returnx is more than 2%, would that tend to shake yourconfidence in the statement that ? = 1.4%? If thishappened, do you think the European stock market might be heatingup? (Round your answer to four decimalplaces.) P(x > 2%)
Explain.
