Sample Statistics Pro ciency Exam #1

Transcription

1 Sample Statistics Pro ciency Exam #1 Name: 1 An appliance store recorded its monthly sales M of microwave ovens for 20 months and ordered them as follows: 123, 126, 140, 141, , 152, 160, 164, , 179, 183, 183, , 210, 231, 233, 274 Summary statistics: mean = 176; standard deviation = 38.5 The pro t is $50 each, less a xed overhead cost of $2,000 per month that is, 50M The mean monthly pro t is therefore: (a) $760 (b) $6,800 (c) $8,800 (d)cannotbedetermined 2WhenisPr(A and B) =Pr(A)Pr(B)? (Let us rule out trivial events with probability of 0 or 1.) (a) Always (b) Never (c) Only if A and B are mutually exclusive (d) Only if A and B are statistically independent 3 Consider the following probability distribution: x Pr(x) The mean of X is: (a) 0.5 (b) 1.0 (c) 0.2 (d) You have four pairs of data points for X and Y : average st. dev. X Y The correlation ½ is: (a) :94 (b) :69 (c) 2:83 (d) :05 1

2 5 A sample of 100 student summer incomes had a mean of $2600, a median of $2100, and a mode of $2000. Then the total income from these 100 students is approximately: (a) $210,000 (b) $260,000 (c) $31,000 (d) need more data 6 The top ve American magazines in the late 1980 s had the following circulation (in millions): Readers Digest 16.4 TV Guide 16.3 National Geographic 10.6 Family Circle 5.9 Woman s Day 5.6 The median of these top ve is, approximately: (a) 16 (b) (c) 11.0 (d) X is normally distributed with a mean of 50 and standard deviation of 10. If Y =3X 2, wecan conclude that Y has a: (a) standard deviation of 90 (b)meanof150 (c) normal distribution (d) standard deviation of 28 8 Weather records for a city indicate that 20% of the days are cloudy, 60% are windy, and 10% are both cloudy and windy. If it is known that a certain day was cloudy, what is the probability that it was windy? (a).33 (b).50 (c).17 (d).10 9 A mail-order catalog has a problem with its service. It nds the time T 1 for an order to arrive in the mail, and the time T 2 for the product to be delivered to the customer, vary according to the following joint distribution (in days): t 2 t Pr(t 1 ) Pr(t 2 ) It takes 2 days to process the order, so that total turnaround time for the customer is T 1 +2+T 2 : This is days on average. 2

3 (a) 6.0 (b) 6.8 (c) 6.2 (d) You are missing an important document, and have forgotten whether it was mislaid at your o ce or at home. But you guess it is twice as likely to be in your o ce. You also estimate, on the basis of past experience, that an initial search in the o ce would have a 60% chance of nding it (if it were there), while an initial search at home would only have a 30% chance. You therefore make the rst search at the o ce. You don t nd it. Now what is the chance that you will nditonyour rstsearchathome? (a).39 (b).17 (c).20 (d) A small piece of hose in the cooling system of a new engine has a lifetime that varies normally around a mean of 18 months, with a standard deviation of 4 months. The rst regular maintenance check occurs at 12 months. The chance the hose will wear out before the maintenance check is: (a).933 (b).067 (c).251 (d) The joint distribution of X and Y is: y x The covariance of X and Y is: (a) 400 (b) 440 (c) 400 (d) Two samples gave the following statistics: rst second sample size mean 4 8 median 3 5 standard deviation 3 3 If combined into one overall sample of 40 observations, the mean is: (a) 7 3

4 (b) 5 (c) 6 (d) 4 14 The chance that a 60 year-old man will die within 10 years is about.26. He can protect his family against this loss by buying $10,000 worth of insurance (10 year term insurance), an agreement by the insurance company to pay his estate $10,000 if he dies within the decade. If the insurance company wants to break even in the long run (i.e., a fair bet, without paying for administrative cost, pro t, or interest, etc.), how much should they charge him at the beginning (at age 60)? (a) $2,600 (b) $7,500 (c) $3,500 (d) $6, The time U it takes Kim Jones to drive to work each day (in minutes) and the time V it takes her to return are random variables whose joint distribution is tabulated below. u v Pr(v) Pr(u) What percent of the time is her total driving time longer than minutes? (a) 50% (b) 95% (c) 65% (d) 80% 16 The average length of human lives is often called life expectancy, and is 73 years (in the US). This means that: (a) more people die at age 73 than at any other age. (b) we can expect to die at about age 73 in the sense that most people die at age 73, give or take a year or two. (c) the mean length of life is 73 years. (d) the median length of life is 73 years. 17 The covariance of X and Y must be zero whenever: (a) X and Y are dependent. (b) both are symmetrically distributed. (c) X and Y are always positive. (d) X and Y are independent. 4

5 18 If E and F are mutually exclusive events with probabilities of.60 and.20, respectively, then the probability of both E and F occurring is: (a).80 (b).68 (c) 0 (d) Joint distribution of X and Y is: y x The mean of Y is: (a) 4 (b) 12 (c) 10 (d) Themedianofasampleofn observations (when n is even) is the: (a) 75th percentile minus the 25th percentile. (b)mostfrequentvalue. (c) middle observation that has (n=2) 1 observations on each side. (d) average of the middle two observations. 21 Consider the following probability distribution: x Pr(x) The standard deviation of X is: (a) 0.4 (b) 0.2 (c) 1.0 (d) Suppose X and Y are independent and identically distributed, so that they have identical means and variances. What is their correlation? (a) 0 (b) +1 (c) 1 (d)cannotbedetermined 5

6 23 For a sample of n =200observations, the relative frequencies were computed as follows: x frequency/n illegible (co ee stain) The missing relative frequency is: (a).15 (b).60 (c).40 (d) impossible to determine 24 An auto manufacturer is trying to foresee some of the problems that will occur in assembling the front wheels of their latest model. For the di erential gear set, three parts have to t into a gap that is mm wide. These three parts are manufactured to a high precision, as follows: Mean Width Standard Deviation Distribution Shape washers 3mm.12 mm normal gears 18 mm.20 mm normal clips 2mm.08 mm normal In assembling a gear set, the three parts will be just randomly drawn from three bins (no attempt will be made to trade o a narrow washer with a wide clip, for example). To see whether they will t into the mm space allotted, we need to know the standard deviation of the combined thickness of all three. It is: (a).40 mm (b).13 mm (c).06 mm (d).25 mm 25 The pro ts that Central Auto makes on its new car sales vary, since some customers drive harder bargains than others. For their medium-sized car, the pro ts are somewhat less than for their fullsized car, as the following tables show: Medium Cars Full-Size Cars pro t $500 (0 1000).50 $1500 ( ).40 $2500 ( ).10 rel. freq. pro t $500 (0 1000).30 $1500 ( ).50 $2500 ( ).20 rel. freq. mean = $1100 mean = $1400 median = $1000 median = $1380 mode = $500 mode = $1500 standard deviation = $663 standard deviation = $700 In opening a new lot in the east end of town, e ciency considerations require them to sell just medium, or just full-size, but not a mix. Since medium cars are easier to sell, they project that for the same outlay of capital and labor, every year they could sell 1000 medium cars compared to 700 full-size cars. To maximize pro t, therefore, they should choose to sell: (a) full-size cars, because their median pro t is larger ($1380 vs. $1000). (b) full-size cars, because their average pro t is larger ($1400 vs. $1100). 6

7 (c) medium cars, because their total pro t is larger ($1,000,000 vs. $966,000 annually). (d) medium cars, because their total pro t is larger ($1,100,000 vs. $980,000 annually). 7