In Lake Erie, walleye fish are abundant. The lengths of walleyes follow a Normal distribution with a mean of 31 inches and a standard deviation of 3.6 inches. Based on this information, which of the following statements is true?a. Approximately 32% of walleyes have lengths less than 27.4 inches.b. Approximately 95% of walleyes have lengths greater than 23.8 inches.c. Approximately 16% of walleyes have lengths longer than 38.2 inches.d. Approximately 84% of walleyes have lengths less than 34.6 inches.d. Approximately 2.5% of walleyes have lengths between 31 inches and 34.6 inches.

d. Approximately 84% of walleyes have lengths less than 34.6 inches.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 31

Standard deviation = 3.6

a. Approximately 32% of walleyes have lengths less than 27.4 inches.

27.4 is one standard deviation below the mean.

By the Empirical Rule, 68% of the measures are within 1 standard deviation of the mean. Of the other 32% more than 1 standard deviation from the mean, 16% of the lengths are less than 27.4 and 16% are greather than 31+3.6 = 34.6.

So approximately 16% of walleyes have lengths less than 27.4 inches, which means that this statement is false.

b. Approximately 95% of walleyes have lengths greater than 23.8 inches.

23.8 is two standard deviations below the mean.

By the Empirical Rule, 95% of the measures are within 2 standard deviations of the mean. Of the 5% that is more than 2 standard deviations from the mean, 2.5% is greater than 2 standard deviations above the mean(greater than 38.2, in this problem), and 2.5% is lower than 2 standard deviations below the mean(lower than 23.8 in this problem).

So 2.5% of walleyes have lengths lower than 23.8 and 100-2.5 = 97.5% have lengths greater than 23.8, which means that this statement is false.

c. Approximately 16% of walleyes have lengths longer than 38.2 inches.

38.2 is two standard deviations above the mean.

By the Empirical Rule, 95% of the measures are within 2 standard deviations of the mean. Of the 5% that is more than 2 standard deviations from the mean, 2.5% is greater than 2 standard deviations above the mean(greater than 38.2, in this problem), and 2.5% is lower than 2 standard deviations below the mean(lower than 23.8 in this problem).

So 2.5% of walleyes have lengths longer than 38.2 inches, which means that this statement is false.

d. Approximately 84% of walleyes have lengths less than 34.6 inches.

34.6 is one standard deviation above the mean.

The empirical rule is symmetric, which means that 50% of the measures are above the mean and 50% of the meausures are below the mean.

Also, 68% of the measures are within 1 standard deviation of the mean. Of those, 34% are between one standard deviation below the mean and the mean, and 34% are between the mean and one standard deviation above the mean.

So, 50% below the mean, plus 34% between the mean and 34.6 means that approximately 84% of the walleyes have lengths less than 34.6 inches, which means that this statement is true.

e. Approximately 2.5% of walleyes have lengths between 31 inches and 34.6 inches.

31 is the mean

34.6 is one standard deviation above the mean.

By the Empirical Rule, 68% of the measures are within 1 standard deviation of the mean. Of those, 34% are between one standard deviation below the mean and the mean, and 34% are between the mean and one standard deviation above the mean.

So 34% of walleyes have lengths between 31 inches and 34.6 inches, which means that this statement is false.