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1.2 - Random and Systematic Errors
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Terms in this set (6)
Error
The difference between a measurement and its accepted value.
Random error
This type of error only affects some of the results. It can be a result of a momentary lapse by the experimenter or because an object's dimensions vary from place to place. Repeat readings can be used to reduce random errors.
Systematic error
This type of error affects all of the data by shifting it either too high or low. This is an error that is built into the apparatus and can not be reduced by repeat readings. To remove the size of the error is found and then minused from all of the results.
Human error
This may be systematic or random, but originates from a mistake that the experimenter did.
Precision
An instrument has a greater precision when it has smaller scale divisions. All readings should be given to the precision of the instrument.
Accurate
A measurement is accurate if it is close to the true value. This can be improved by repeat readings.
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Verified questions
STATISTICS
Suppose that a manufacturing process makes about 3% defective items, which is considered satisfactory for this particular product. The managers would like to decrease this to about 1% and clearly want to guard against a substantial increase, say to 5%. To monitor the process, periodically n = 100 items are taken and the number X of defectives counted. Assume that X is $b ( n = 100 , p = \theta ).$ Based on a sequence $X_1,X_2, . . . , X_m, . . .,$ determine a sequential probability ratio test that tests $H _ { 0 } : \theta = 0.01$ against $H _ { 1 } : \theta = 0.05.$ (Note that $\theta = 0.03$, the present level, is in between these two values.) Write this test in the form $h _ { 0 } > \sum _ { i = 1 } ^ { m } \left( x _ { i } - n d \right) > h _ { 1 }$ and determine d $h _ { 0 } ,$ and $h _ { 1 }$ if $\alpha _ { a } = \beta _ { a } = 0.02$
PROBABILITY
A new radar device is being considered for a certain missile defense system. The system is checked by experimenting with aircraft in which a kill or a no kill is simulated. If, in 300 trials, 250 kills occur, accept or reject, at the 0.04 level of signiﬁcance, the claim that the probability of a kill with the new system does not exceed the 0.8 probability of the existing device.
PROBABILITY
The ﬁnancial structure of a ﬁrm refers to the way the ﬁrm’s assets are divided into equity and debt, and the ﬁnancial leverage refers to the percentage of assets ﬁnanced by debt. In the paper The Effect of Financial Leverage on Return, Tai Ma of Virginia Tech claims that ﬁnancial leverage can be used to increase the rate of return on equity. To say it another way, stockholders can receive higher returns on equity with the same amount of investment through the use of ﬁnancial leverage. The following data show the rates of return on equity using 3 different levels of ﬁnancial leverage and a control level (zero debt) for 24 randomly selected ﬁrms: $$ \begin{matrix} & \text{Financial Leverage }\\ \text{Control} & \text{Low} & \text{Medium} & \text{High}\\ \text{2.1} & \text{6.2} & \text{9.6} & \text{10.3}\\ \text{5.6} & \text{4.0} & \text{8.0} & \text{6.9}\\ \text{3.0} & \text{8.4} & \text{5.5} & \text{7.8}\\ \text{7.8} & \text{2.8} & \text{12.6} & \text{5.8}\\ \text{5.2} & \text{4.2} & \text{7.0} & \text{7.2}\\ \text{2.6} & \text{5.0} & \text{7.8} & \text{12.0}\\ \end{matrix} $$ Source: Standard & Poor’s Machinery Industry Survey, 1975. (a) Perform the analysis of variance at the 0.05 level of signiﬁcance. (b) Use Dunnett’s test at the 0.01 level of signiﬁcance to determine whether the mean rates of return on equity are higher at the low, medium, and high levels of ﬁnancial leverage than at the control level.
PROBABILITY
In an endless soccer match, goals are scored according to a Poisson process with rate $\lambda$. Each goal is made by team A with probability p and team B with probability 1−p. For j > 1, we say that the jth goal is a turnaround if it is made by a different team than the (j − 1)st goal; for example, in the sequence AABBA. . . , the 3rd and 5th goals are turnarounds. (a) In n goals, what is the expected number of turnarounds? (b) What is the expected time between turnarounds, in continuous time?