Question

The weights of the fish in a certain lake are normally distributed with a mean of 19 lb and a standard deviation of 6. If 4 fish are randomly selected, what is the probability that the mean weight will be between 16.6 and 22.6 lb

Answer 1

Answer:

67.30% probability that the mean weight will be between 16.6 and 22.6 lb

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X

Central limit theorem:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 19, \sigma = 6, n = 4, s = \frac{6}{\sqrt{4}} = 3$

If 4 fish are randomly selected, what is the probability that the mean weight will be between 16.6 and 22.6 lb

This is the pvalue of Z when X = 22.6 subtracted by the pvalue of Z when X = 16.6.

X = 22.6

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{22.6 - 19}{3}$

$Z = 1.2$

$Z = 1.2$ has a pvalue of 0.8849

X = 16.6

$Z = \frac{X - \mu}{s}$

$Z = \frac{16.6 - 19}{3}$

$Z = -0.8$

$Z = -0.8$ has a pvalue of 0.2119

0.8849 - 0.2119 = 0.6730

67.30% probability that the mean weight will be between 16.6 and 22.6 lb