Calculating Probabilities

[oldoc63]

Once we know the sampling distribution of the mean, we can also use to estimate the probability of observing a particular range of sample means, given some information (either known or assumed) about the population. To do this, we can use the Cumulative Distribution Function, or CDF of the normal distribution.

Let's work throuhg this with our salmon fish example. Let's say we are transporting the salmon and want to make sure the crate we carry the fish in will be strong enough to hold the weight.

• Suppose we estimate that the salmon population has an average weigh of 60 lbs with a standard deviation of 40 lbs.
• We have a crate that supports 750 lbs, and we want to be able to transport 10 fish at a time.
• We want to calculate the probability that the average weight of those 10 fish is less than or equal to 75 (750/10).

Using the CLT, we first estimate that the mean weight of 10 randomly sampled salmon from this population is normaly distributed with mean = 60 and standard error = 40/10^.5. Then we can use this probability distribution to calculate the probability that 10 ramdomly sampled fish will have a mean weight less than or equal to 75.

[oldoc63]

Let's calculate the same probability for our cod fish poupulation.

• We know that cod have an average weight of 36 lbs with a standard deviation of 20.
• We want to try to fit 25 cod fish into our same crate that can hold up to 750 lbs.

Our first step is to calculate the standard error for a sample size of 25. Then use the normal CDF to calculate the probability that a sample of 25 fish has a mean weight of 30 lbs.