4.   The data set of Exercise 1 also includes these third- grade verbal IQ observations for males:

Let

 

 

Y = sample mean book value

X = sample mean audited value

D = sample mean error

 

 

117   103   121	112	120	132	113	117	132
149   125   131	136	107	108	113	136	114
and females:	Â	Â	Â	Â	Â	Â
114   102   113	131	124	117	120	90	Â
114   109   102	114	127	127	103	Â	Â

Prior to obtaining data, denote the male values by X1,... , Xm and the female values by Y1,... , Yn. Sup- pose that the Xi s constitute a random sample from a distribution with mean m1 and standard deviation s1 and that the Yi s form a random sample (independent of the Xi  s) from another distribution with mean m2

Several different statistics for estimating the total audited (correct) value have been proposed (see Statistical Models and Analysis in Auditing, Sta- tistical Sci., 1989: 2—33). These include

 

Mean per unit statistic = NX    Difference statistic = T  - ND Ratio statistic = T # 1X/Y 2

If N = 5000 and T = 1,761,300, calculate the three corresponding point estimates. (The cited article dis- cusses properties of these estimators.)

 

and standard deviation s2.

                     6.  Consider the accompanying observations on stream

 

a.   Use rules of expected value to show that X - Y is an unbiased estimator of m1 - m2. Calculate the estimate for the given data.

b.   Use rules of variance from Chapter 6 to obtain an

ow  (1000 s of acre-feet) recorded at a station    in

Colorado for the period April 1—August 31 over a 31-year span (from an article in the 1974 volume of Water Resources Res.).

 

 

expression for the variance and standard  devia-	127.96	210.07	203.24	108.91	178.21
tion (standard error) of the estimator in part  (a),	285.37	100.85	89.59	185.36	126.94
and then compute the estimated standard   error.	200.19	66.24	247.11	299.87	109.64

 

125.86	114.79	109.11	330.33	85.54	Â	ity that a component works properly, how    can
117.64	302.74	280.55	145.11	95.36	Â	P(system works) be expressed in terms of p?]
204.91	311.13	150.58	262.09	477.08	c.	Let  pˆ         be the sample proportion of successes. Is  pˆ    2
94.33	Â	Â	Â	Â	Â	an unbiased estimator for p2? Hint: For any rv Y, E(Y 2) = V(Y) + [E(Y)]2.

 

An appropriate probability plot supports the use of the lognormal distribution (see Section 4.5) as a rea- sonable model for stream   ow.

a.   Estimate the parameters of the distribution. [Hint: Remember that X has a lognormal distri- bution with parameters m and s2 if ln(X) is nor- mally distributed with mean m and variance s2.]

b.   Use the estimates of part (a) to calculate an esti- mate of the expected value of stream ow. [Hint: What is E(X)?]