Question

A large tank of fish from a hatchery is being delivered to a lake. The hatchery claims that the mean length of fish in the tank is 15 inches, and the standard deviation is 7 inches. A random sample of 53 fish is taken from the tank. Let x be the mean sample length of these fish. What is the probability that x is within 0.5 inch of the claimed population mean? (Round your answer to four decimal places.)

Answer 1

Answer:

Given Data

Mean = 15

SD = 7

Now , no of sample = 53

Sample SD

$=\frac{SD}{\sqrt{n}}$

$=\frac{7}{\sqrt{53}}$

= 0.9615

Let the sample mean be x

X is normally distributed with a mean 15 and SD = 0.9615

Converting to standard normal,

$z = \frac{(x-15)}{0.9615}$

P ( x lies within 0.5 inch of population mean )

$=P((14.5-15)/0.9615<z<(15.5-15)/0.9615)$

$=P(-0.52<z<0.52)$

$=P(z<0.52)-P(z<-0.52)$

$=P(z<0.52)-[1-P(z<0.52)]$

$=2P(z<0.52)-1$

$=2(0.69847)-1$

= 0.39694

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