In a survey conducted by the Big Creek State Marine Reserve

In a survey conducted by the Big Creek State Marine Reserve in California, the length of various species of fish caught at two locations were recorded. Suppose the length of a yellowtail caught at the Santa Lucia Kelp Bed is normally distributed with mean 38 inches and standard deviation 2.9 inches.


1. The probability that the length of one yellowtail selected at random is more than 41 inches is _(Answer 1)_.

2. Suppose 32 yellowtails are selected at random. The probability that the sample mean length is less than 38.5 inches is _(Answer 2)_.

3. Suppose 32 yellowtails are selected at random. The probability that the sample mean is between 37 and 40 inches is _(Answer 3)_.

Solution

(1) P(X>41) = P((X-mean)/s >(41-38)/2.9)

=P(Z>1.03) = 0.1515 (from standard normal table)

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(2) P(xbar<38.5) = P((xbar-mean)/(s/vn) <(38.5-38)/(2.9/sqrt(32)))

=P(Z<0.98) =0.8365 (from standard normal table)

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(3) P(37 <xbar< 40) = P((37-38)/(2.9/sqrt(32)) <Z< (40-38)/(2.9/sqrt(32)))

=P(-1.95<Z<3.90)

=0.9744 (from standard normal table)