Question

A bottling plant fills one-gallon jugs with milk. The labels on the jugs state that the volume of milk they contain is 128 fluid ounces (fl. oz.). Federal law mandates that the jugs must contain no less than the stated volume. The actual volume of milk in the jugs is normally distributed with mean μ=129 fl. oz. and standard deviation σ=0.8 fl. oz. Plant workers take a simple random sample (SRS) of 8 jugs, measure the volume of milk in each jug, and calculate the sample mean.
Use this information to answer questions1. What is the shape of the distribution of sample means?A. Skewed rightB. NormalC. Skewed left2. Calculate the mean of the distribution of sample means. Round your answer to the nearest whole number. 3. Calculate the standard deviation of the distribution of sample means. Round to three decimal places4. If plant workers sampled 100 jugs instead of 8, which one of the following changes in the distribution of means would they see?A. The mean would decrease.B. The standard deviation would increase.C. The standard deviation would decrease.D. The sample means would be uniformly distributed.

Answer 1

Answer:

(1) The correct option is (B).

(2) The mean of the distribution of sample means is 19 fl. oz.

(3) The standard deviation of the distribution of sample means is 0.283 fl. oz.

(4) The correct option is (C).

Step-by-step explanation:

According to the Central Limit Theorem if we have a normal population with mean μ and standard deviation σ and a number of random samples are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

The mean of the sampling distribution of sample mean will be:

$\mu_{\bar x}=\mu$

And the standard deviation of the sampling distribution of sample mean will be:

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

The information provided is:

μ = 129

σ = 0.80

n = 8

(1)

The shape of the sampling distribution of sample mean will be Normal.

This is because the population from which the sample is selected is normal.

The correct option is (B).

(2)

Compute the mean of the distribution of sample means as follows:

$\mu_{\bar x}=\mu$

    $=129$

Thus, the mean of the distribution of sample means is 19 fl. oz.

(3)

Compute the standard deviation of the distribution of sample means as follows:

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

    $=\frac{0.80}{\sqrt{8}}\\\\=0.282843\\\\\approx 0.283$

Thus, the standard deviation of the distribution of sample means is 0.283 fl. oz.

(4)

Any change in the sample size will have no effect on the mean of the distribution of sample means.

But, if the sample is increased or decreased than the standard deviation will be decreased or increased respectively.

This is because the standard deviation of the distribution of sample means is inversely proportional to the sample size.

So, for n = 100 the standard deviation is:

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

    $=\frac{0.80}{\sqrt{10}}\\\\=0.282843\\\\\approx 0.08$

Thus, the standard deviation was decreased.

The correct option is (C).

Answer 2

Using the Central Limit Theorem, it is found that:

1. B. Normal.

2. The mean is of 129 fl. oz.

3. The standard deviation is of 0.283.

4. C. The standard deviation would decrease.

Central Limit Theorem

• The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n is approximately normal with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

• For a skewed variable, the sampling distribution is also approximately normal, as long as n is at least 30.

Item a:

• The underlying distribution is normal, hence the distribution is normal and option B is correct.

Item b:

• The mean of the population is of 129 fl. oz, hence it is the same for the distribution of sample means.

Item c:

• The population standard deviation is of $\sigma = 0.8$.

• A sample of size 8 is taken, hence $n = 8$.

Then:

$s = \frac{0.8}{\sqrt{8}} = 0.283$

Item d:

The standard deviation is inversely proportional to the square root of the sample size, hence it would decrease, and option C is correct.

To learn more about the Central Limit Theorem, you can take a look at https://brainly.com/question/24663213