Question

The size of fish is very important to commercial fishing. A study conducted in 2012 found the length of Atlantic cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm (Ovegard, Berndt & Lunneryd, 2012). Assume the length of fish is normally distributed. Round the probabilities to four decimal places. It is possible with rounding for a probability to be 0.0000. a.) Find the probability that an Atlantic cod has a length that is less than 58.98 cm. b.) Find the probability that an Atlantic cod has a length that is between 58.88 and 58.98 cm. c.) Find the probability that an Atlantic cod has a length that is at least 55.51 cm. d.) Is a length of 55.51 cm unusually long for an Atlantic cod? Why or why not? yes, since the probability of having a length at least that high is less than or equal to 0.05 yes, since the probability of having a length at the most that value is less than or equal to 0.05 no, since the probability of having a length at least that high is less than or equal to 0.05 no, since the probability of having a length at the most that value is less than or equal to 0.05 yes, since the probability of having a length at least that high is greater than 0.05 yes, since the probability of having a length at the most that value is greater than 0.05 no, since the probability of having a length at least that high is greater than 0.05 no, since the probability of having a length at the most that value is greater than 0.05 ОО e.) What lengths are 65% of all Atlantic cod longer than? Put an answer rounded to two decimal places in the first box and the correct units in the second box.

Answer 1

Solution: given that mean = 49.9, sd = 3.74
a) P(X < 58.98) = P((X-mean)/sd < (58.98-49.9)/3.74)
= P(Z < 2.4278)
= 0.9925

b) P(58.88 < X < 58.98) = P((58.88-49.9)/3.74 < Z < (58.98-49.9)/3.74)
= P(2.4011 < Z < 2.4278)
= P(Z < 2.4278) - P(Z < 2.4011)
= 0.9925 - 0.9918
= 0.0007

c) P(X > 55.51) = P(Z > (55.51-49.9)/3.74)
= P(Z > 1.5)
= 0.0668

d) => option g.

e) for P(X > x) = 0.65 => Z = -0.385

=> X = 49.9 - (0.385*3.74) = 48.46