Q3: Although errors are likely when taking measurements from photographic images
ID: 3336873 • Letter: Q
Question
Q3: Although errors are likely when taking measurements from photographic images, these errors are often very small For sharp images with negligible distortion, errors in measuring distances are often no larger than 0.0004 inches. Assume that the probability of a serious measurement error is o independent measurements are make. Let X denote the number of serious errors made .What kind of distribution is this describing? Is using the normal approximation appropriate? Why or why not? · Find the probability of making at least one serious error .Find the probability of making no more than 3 serious errors Find the probability of making exactly 3 serious errors Suppose you are a sloppy measuretlfyou are an the top 10% oferor makers, what s the minimum number of errors you could have made? .Explanation / Answer
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
mean ( np ) = 150 * 0.05 = 7.5
standard deviation ( npq )= 150*0.05*0.95 = 2.6693
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)
a. normal distribution
b.
Points to pass for normal approximation:
1) experiment consistes of a sequence of n identical trials
2) only 2 outcomes are possible on each trail, success or failure
3) trials are independent & below conditions should satisfy
n*p>5, 150*0.05> 5 => 7.5>5
n*(1-p)>5, 150*0.95> 5 => 7.5>5
can use normal approximation
c.
P(X < 1) = (1-7.5)/2.6693
= -6.5/2.6693= -2.4351
= P ( Z <-2.4351) From Standard NOrmal Table
= 0.0074
P(X > = 1) = (1 - P(X < 1))
= 1 - 0.0074 = 0.9926
d.
P(X > 3) = (3-7.5)/2.6693
= -4.5/2.6693 = -1.6858
= P ( Z >-1.6858) From Standard Normal Table
= 0.9541
P(X < = 3) = (1 - P(X > 3))
= 1 - 0.0459 = 0.9541
e.
P( X = 3 ) = ( 150 3 ) * ( 0.05^3) * ( 1 - 0.05 )^147
= 0.0366
f.
P ( Z > x ) = 0.1
Value of z to the cumulative probability of 0.1 from normal table is 1.2816
P( x-u / (s.d) > x - 0.05/150) = 0.1
That is, ( x - 0.05/150) = 1.2816
--> x = 1.2816 * 150+0.05 = 192.2827
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