A particular quantity is measured 500 times. The results are observed to have a

Question

A particular quantity is measured 500 times. The results are observed to have a roughly Gaussian distribution, with a mean x = 89.5 units and a standard deviation s_x = 12.5units. a. What is the probability that the next measurement will be in the range 77-102 units? b. What is the probability that the next measurement will exceed 100 units? c. Estimate the range in which 95% of any individual measurements will fall. d. Estimate the 95% confidence interval for the mean mu of the population of all possible measurements

Explanation / Answer

Normal Distribution
Mean ( u ) =89.5
Standard Deviation ( sd )=12.5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a.
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 77) = (77-89.5)/12.5
= -12.5/12.5 = -1
= P ( Z <-1) From Standard Normal Table
= 0.15866
P(X < 102) = (102-89.5)/12.5
= 12.5/12.5 = 1
= P ( Z <1) From Standard Normal Table
= 0.84134
P(77 < X < 102) = 0.84134-0.15866 = 0.6827                  

b.
P(X > 100) = (100-89.5)/12.5
= 10.5/12.5 = 0.84
= P ( Z >0.84) From Standard Normal Table
= 0.2005                  

c.
About 95% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 2s.d)
So to the given normal distribution about 95% of the observations lie in between
= [89.5 ± 2 * 12.5]
= [ 89.5 - 2 * 12.5 , 89.5 + 2* 12.5]
= [ 64.5 , 114.5 ]  

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