The distribution of actual weights of 8-ounce wedges of cheddar cheese produced at a dairy is Normal with mean 8.1 ounces and standard deviation 0.2 ounces. A sample of 10 of these cheese wedges is selected. The distribution of the sample mean of the weights of cheese wedges is: approximately Normal, mean 8.1, standard deviation 0.2. approximately Normal, mean 8.1, standard deviation 0.020. It is not possible to tell because the sample size is too small. approximately Normal, mean 8.1, standard deviation 0.063. 15. The distribution of actual weights of 8-ounce wedges of cheddar cheese produced at a dairy is Normal with mean 8.1 ounces and standard deviation 0.2 ounces. A sample of 10 of these cheese wedges is selected. What is the standard deviation of the sampling distribution of the mean? 0.075 ounces 0.963 ounces 0.315 ounces 0.0633 ounces 16. The distribution of actual weights of 8-ounce wedges of cheddar cheese produced at a dairy is Normal with mean 8.1 ounces and standard deviation 0.2 ounces. A sample of 10 of these cheese wedges is selected. The company decides instead to sample batches of 20 cheese wedges, and the sampling is repeated every time workers start a new shift at the dairy. How will the distribution of the sample means of the weights of cheese wedges change from the previous batches, which only contained 10 samples? The distribution will still be Normal, but it will be more peaked around the sample mean and the standard deviation will be larger. The distribution will still be Normal, but it will be more peaked around the sample mean and the standard deviation will be smaller. The shape of the distribution may change completely based on the new data. It is not possible to tell from the information provided.

14. The distribution of the sample mean for the weights of the cheese wedges follows a Normal distribution with the same mean as the population mean, which is 8.1 ounces. However, the standard deviation of the sample mean (often called the standard error) is calculated by dividing the population standard deviation (σ = 0.2 ounces) by the square root of the sample size (n). Therefore, the standard deviation of the sample mean for samples of size 10 is:

Standard deviation of the sample mean = σ/√n = 0.2/√10 ≈ 0.0633 ounces.

Thus, the correct answer for the distribution of the sample mean is approximately Normal, mean 8.1, standard deviation 0.0633 ounces.

15. For the standard deviation of the sampling distribution of the mean, also known as the standard error, we use the same calculation as above, as we have the population standard deviation and the sample size:

Standard deviation of the sample mean (Standard Error) = σ/√n = 0.2/√10 ≈ 0.0633 ounces.

Therefore, the standard deviation of the sampling distribution of the mean is 0.0633 ounces.

16. When the sample size increases from 10 to 20, the standard deviation of the sampling distribution of the mean decreases. This is because the standard error of the mean is inversely related to the square root of the sample size. As the sample size increases, the distribution of the sample means will become more narrowly clustered around the population mean because there is less variability in larger samples. The formula for the standard error is:

Standard deviation of the sample mean = σ/√n.

Using a sample size of 20, we get:

Standard deviation of the sample mean = 0.2/√20 < 0.2/√10.

This shows that the standard deviation with a sample size of 20 is smaller than when the sample size is 10. Therefore, the distribution will still be Normal, the standard deviation will be smaller, and it will be more peaked around the sample mean.

Extra: When discussing distributions and sample means, it's important to understand the Central Limit Theorem. This theorem states that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution, provided the sample size is sufficiently large (usually n > 30 is considered large enough).

The mean of the sampling distribution of the sample mean is equal to the population mean. This concept is key in inferential statistics as it allows us to make estimations about the population from samples.

The standard deviation of the sampling distribution of the sample mean (standard error) tells us how much we can expect sample means to vary from the population mean. A smaller standard error means that most sample means are close to the population mean, which indicates that our estimate of the population mean based on the sample mean is likely to be accurate.

When increasing the sample size, the variability of the sample means (the standard error) decreases. This makes the sampling distribution narrower, which means that our sample means are getting closer to the true population mean. The larger the sample, the more confident we can be in our estimates of the population parameters.