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

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 and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sample means with size n can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

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

By the Central Limit Theorem

has a pvalue of 0.8849

X = 16.6

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

step-by-step explanation:

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