Answer:
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.
