Question

Nonpoint source loads are chemical masses that travel to the main stem of a river in flows that are distributed over relatively long stream reaches. Suppose source loads are known to have a mean value of 73 and a standard deviation of 10. We will take a random sample of 48 loads and calculate the sample mean, X. (a) What is the mean of X? (b) What is the variance of X? (c) Sketch the approximate probability density function of X (roughly). It does not need to be exact, but it should be centered in the right place and have the correct basic spread and shape. (d) Find the (approximate) probability that the sample mean load is between 71.91 and 73.08, inclusive. (e) If the sample size had been 17 instead of 48, could the probability in the previous part be calculated from the given information? Explain

Answer 1

a)

Mean of X = 73

b)

Standard deviation of X = 10 / sqrt(48) = 1.443375673

Variance = S.D ^ S.D = 1.443375673 ^2 = 2.083

c)

µ =    73                                  
σ =    1.44

d)

µ =    73                                  
σ =    10                                  
n=   48                                  
we need to calculate probability for ,                                      
71.91   ≤ X ≤    73.08                              
X1 =    71.91   ,    X2 =   73.08                      
                                      
Z1 =   (X1 - µ )/(σ/√n) = (   71.91   -   73   ) / (   10   / √   48   ) =   -0.76
Z2 =   (X2 - µ )/(σ/√n) = (   73.08   -   73   ) / (   10   / √   48   ) =   0.06
                                      
P (   71.91   < X <    73.08   ) =    P (    -0.8   < Z <    0.1   )   
                                      
= P ( Z <    0.06   ) - P ( Z <   -0.76   ) =    0.52210   -    0.22507   =    0.2970  

e)
As a general rule, sample sizes equal to or greater than 30 are deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed.

However if sample size is lesser than 30 and pop15ulation is not normal than CLT will not hold