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Is it possible to estimate the uncertainty in this measurement?

1.  For some measuring processes, the uncertainty is approximately proportional to the value of the measurement. For example, a certain scale is said to have an uncertainty of ±2%. An object is weighed on this scale.

 

a.   Given that the reading is 100 g, express the uncertainty in this measurement in grams.

b.  Given that the reading is 50 g, express the uncertainty in this measurement in grams.

 

2.   A person stands on a bathroom scale. The reading is 150 lb. After the person gets off the scale, the reading is 2 lb.

a.  Is it possible to estimate the uncertainty in this measurement?

b.  If so, estimate it. If not, explain why not. Is it possible to estimate the bias in this measurement? If so,….

Is it possible to estimate the bias in these measurements?

1.   A person gets on and off a bathroom scale four times. The four readings (in pounds) are 148, 151, 150, and 152. Each time after the person gets off the scale, the reading is 2 lb.

 

a.  Is it possible to estimate the uncertainty in these measurements? If so, estimate it. If not, explain why not.

b.  Is it possible to estimate the bias in these measurements? If so, estimate it. If not, explain why not.

 

2.   In a hypothetical scenario, the National Institute of Standards and Technology has received a new replica of The Kilogram. It is weighed five times. The measurements are as follows, in units of micro-grams above 1 kg: 114.3, 82.6, 136.4, 126.8, 100.7.

 

a.   Is it possible to estimate….

Is it possible to estimate the uncertainty in these measurements?

1.  The Kilogram is now weighed five times on a different scale. The measurements are as follows, in units of micrograms above 1 kg: 25.6, 26.8, 26.2, 26.8, 25.4.

 

a.   Is it possible to estimate the uncertainty in these measurements? If so, estimate it. If not, explain why not.

b.   Is it possible to estimate the bias in these measurements? If so, estimate it. If not, explain why not.

 

 

2.   A new and unknown weight is weighed on the same scale that was used in Exercise 8, and the measurement is 127 μg above 1 kg. Using the information in Exercise 8, is it possible to come up with a more accurate measurement? If so, what is it? If not, explain why not.

 

3. ….

Find the uncertainties in the following quantities

1.  The length of a rod was measured eight times. The measurements in centimeters, in the order they were taken, were 21.20, 21.22, 21.25, 21.26, 21.28, 21.30, 21.32, 21.35.

a.  Do these measurements appear to be a random sample from a population of possible measurements? Why or why not?

b.  Is it possible to estimate the uncertainty in these measurements? Explain.

 

 

2.   Assume that X and Y are independent measurements with uncertainties σX = 0.3 and σY = 0.2. Find the uncertainties in the following quantities:

a.   4X X + 2Y

b.   2X − 3Y

 

3.   A measurement of the diameter of a disk has an uncertainty of 1.5 mm. How many measurements must be made so that the diameter can be estimated with an….

Estimate the difference in the mean annual uplift between these two time periods, and find the uncertainty in the estimate.

1.  The length of a rod is to be measured by a process whose uncertainty is 3 mm. Several independent measurements will be taken, and the average of these measurements will be used to estimate the length of the rod. How many measurements must be made so that the uncertainty in the average will be 1 mm?

2.   The volume of a cone is given by V = πr 2h/3, where r is the radius of the base and h is the height. Assume the radius is 5 cm, measured with negligible uncertainty, and the height is h = 6.00 ± 0.02 cm. Estimate the volume of the cone, and find the uncertainty in the estimate.

3.   In the article “The World's Longest Continued Series of Sea Level….

Estimate the specific gravity, and find the uncertainty in the estimate.

1.  A cylindrical hole is bored through a steel block, and a cylindrical piston is machined to fit into the hole. The diameter of the hole is 20.00 ± 0.01 cm, and the diameter of the piston is 19.90 ± 0.02 cm. The clearance is one-half the difference between the diameters. Estimate the clearance and find the uncertainty in the estimate.

2.   A force of F = 2.2 ± 0.1 N is applied to a block for a period of time, during which the block moves a distance d = 3 m, which is measured with negligible uncertainty. The work W is given by W = Fd. Estimate W, and find the uncertainty in the estimate.

3.   The specific gravity of a substance is given by G =….

Find the conditional probability mass function pY|X (y | 3).

1.  Refer to Exercise 12.

a.   Let Z = X + Y represent the total number of repairs needed.

b.  Find µZ .

c.    Find σZ .

d.   Find P(Z = 2).

 

2.   Refer to Exercise 12. Assume that the cost of an engine repair is $50, and the cost of a transmission repair is $100. Let T represent the total cost of repairs during a one-hour time interval.

a.  Find µT .

b.  Find σT .

c.    Find P(T = 250)

 

3.  Refer to Exercise 12.

 

a.   Find the conditional probability mass function pY|X (y | 3).

b.   Find the conditional probability mass function pY|X (x | 1).

c.    Find the conditional expectation E(Y | X = 3).

d.  Find the conditional expectation E(X….

Find the value of K that minimizes the risk in part (c).

1.  Refer to Example 2.54.

a.  Find Cov(X,Y).

b.  Find ρX,Y .

 

 

2.    Refer to Exercise 18.

a.  Find Cov(X, Y).

b.   Find ρX,Y .

c.    Find the conditional probability density function fY|X (y | 0.5).

d.  Find the conditional expectation E(Y | X = 0.5).

 

3.   An investor has $100 to invest, and two investments between which to divide it. If she invests the entire amount in the first investment, her return will be X, while if she invests the entire amount in the second investment, her return will be Y. Both X and Y have mean $6 and standard deviation (risk) $3. The correlation between X and Y is 0.3.

 

a.  Express the return in terms of X and Y if she….

Prove that Cov(aX, bY) = ab Cov(X,Y).

1.  Let X and Y be random variables, and a and b be constants.

a.  Prove that Cov(aX, bY) = ab Cov(X,Y).

b.  Prove that if a > 0and b > 0, then ρaX,bY = ρX,Y .

c.   Conclude that the correlation coefficient is unaffected by changes in units.

 

2.   Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z). (Hint: Use Equation 2.71.)

 

 

3.  Let X and Y be jointly distributed random variables. This exercise leads you through a proof of the fact that −1 ≤ ρX,Y ≤ 1.

a.   Express the quantity V(X − (σX /σY )Y) in terms of σx , σY , and Cov(X, Y).

b.  Use the fact….

Use part (a) to show that X and Y are independent.

1.  Refer to Exercise 30. An equation to predict the ductility of a titanium weld is Y = 7.84C + 11.44N + O − 1.58Fe, where Y is the oxygen equivalence used to predict ductility, and C, N, O, and Fe are the amounts of carbon, nitrogen, oxygen, and iron, respectively, in weight percent, in the weld. Using the means, standard deviations, and correlations presented in Exercise 30, find µY and σY

 

2.  Let X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX (x) and fY (y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative.

a.  Show that there….