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A commercial crabber catches more than 1,000 crabs and measures the shells, and finds the mean length is 6.8 inches with a standard deviation of 3.2 inches. Assuming these measures are true for the population, if the crabber takes many random samples of size 50, what proportion of the sample means would we expect to be greater than 6 inches?
A. 0.8815
B. 0.9615
C. 0.0385
D. 0.1848
E. 0.4999

Answer :

JeanaShupp

Answer:  B . 0.9615

Step-by-step explanation:

Step-by-step explanation:

Let x be a random variable that represents the lengths of the shells .

As per given , we have

[tex]\mu=6.8[/tex] inches

[tex]\sigma=3.2[/tex] inches

n= 50

∵ [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Then for x=  6,

[tex]z=\dfrac{6-6.8}{\dfrac{3.2}{\sqrt{50}}}=-1.76776695297\approx-1.7678[/tex]  

The probability the sample means would we expect to be greater than 6 inches :-

[tex]P(x>6)=P(z>-1.7678)=P(z<1.7678)\ \ [\because P(Z>-z)=P(Z<z)]\\\\=0.9614528\approx0.9615[/tex]  [using the z values table]

Hence, the proportion of the sample means would we expect to be greater than 6 inches =  0.9615

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