# 7 Central Limit Theorem

# Problem 1

A can of Ocean brand tuna is supposed to have a net weight of 6 ounces. The manufacturer tells you that the net weight is actually a Normal random variable with a mean of 5.95 ounces and a standard deviation of 0.2 ounces. Suppose that you draw a random sample of 42 cans.

**Part i) **Suppose the number of cans drawn is doubled. How will the standard deviation of sample mean weight change?

- It will increase by a factor of 2.
- It will decrease by a factor of 2.
- It will decrease by a factor of square root of 2.
- It will increase by a factor of square root of 2.
- It will remain unchanged.

**Part ii) **Suppose the number of cans drawn is doubled. How will the mean of the sample mean weight change?

- It will increase by a factor of square root of 2.
- It will increase by a factor of 2.
- It will decrease by a factor of square root of 2.
- It will decrease by a factor of 2.
- It will remain unchanged.

**Part iii) **Consider the statement: ’The distribution of the mean weight of the sampled cans of Ocean brand tuna is Normal.’

- It is a correct statement, but it is not a result of the Central Limit Theorem.
- It is a correct statement, and it is a result of the Central Limit Theorem.
- It is an incorrect statement. The distribution of the mean weight of the sample is not Normal.

## Solution

The distribution of the weight of a single can of tuna is given to be Normal with mean 5.95 oz. and standard deviation 0.2 oz.

(i) The sample mean weight of a sample of 42 such cans is a random variable that follows the Normal distribution with mean 5.95 and standard deviation [latex]\frac{0.2}{\sqrt{42}}[/latex]. If we were to double the sample size to 84, the standard deviation of the sample mean would decrease by a factor of [latex]\sqrt{2}[/latex].

(ii) Changing the sample size does not affect the mean of the sampling distribution of the sample mean, which would remain at 5.95.

(iii) The sample mean here is normally distributed, though there is no need to apply the Central Limit Theorem for this result. Instead, the result follows exactly since the distribution of the weights of the individual cans is Normal.

## Correct Answers

- C
- E
- A