Question 1

The mean of a sampling distribution would be equal to the mean of the population from which the sampling distribution is constructed.

Accepted Answer

The mean of a sampling distribution (also known as the expected value of the sample mean) is equal to the mean of the population from which the samples are drawn, according to the Central Limit Theorem. This property holds as long as the samples are drawn randomly and the population size is much larger than the sample size.

Question 2

The population of soft drink cans filled by a particular machine is known to be normally distributed with a mean equal to 12 ounces and a standard deviation equal to .25 ounce. Given this information, the sampling distribution for a random sample of n = 25 cans will also be normally distributed with a mean equal to 12 ounces and a standard deviation equal to .05 ounce.

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the sample mean will be normally distributed if the population from which the sample is drawn is normally distributed. The mean of the sampling distribution ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$), which is 12 ounces in this case. The standard deviation of the sampling distribution ($\sigma_{\bar{x}}$) is equal to the population standard deviation ($\sigma$) divided by the square root of the sample size ($n$), which calculates to $0.25 / \sqrt{25} = 0.25 / 5 = 0.05$ ounce.

Question 3

A sampling distribution for a sample of n = 4 is normally distributed with a standard deviation equal to 5. Based on this information, the population standard deviation, σ, is equal to 10 mph.

Accepted Answer

The answer of A sampling distribution for a sample of...

Question 4

If you are sampling from a very large population, a doubling of the sample size will reduce the standard error of the sampling distribution by one-fourth.

Accepted Answer

Doubling the sample size reduces the standard error by a factor of $\sqrt{2}$, not by one-fourth. The standard error of the mean is inversely proportional to the square root of the sample size ($n$), so increasing the sample size by a factor of 2 reduces the standard error by a factor of $\sqrt{2}$, which is approximately 1.41, not by one-fourth.

Question 5

The Central Limit Theorem is of most use to decision makers when the population is known to be normally distributed.

Accepted Answer

The Central Limit Theorem is useful because it applies regardless of the population's distribution, stating that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large.

Question 6

Sampling error can be eliminated if the sample size is large enough.

Accepted Answer

Sampling error can decrease with an increase in sample size but cannot be completely eliminated as it is inherent to the process of sampling.

Question 7

A sampling distribution for is the distribution of all possible sample means that could be computed from the possible samples of a given sample size.

Accepted Answer

A sampling distribution is indeed the distribution of all possible sample means (or other statistics) that could be computed from all possible samples of a given size drawn from the same population.

Question 8

One of the things that the Central Limit Theorem tells us is that about half of the sample means will be greater than the population mean and about half will be less.

Accepted Answer

The Central Limit Theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution. In a normal distribution, about half of the values will be greater than the mean and about half will be less.

Question 9

It is very unlikely that a nonstatistical sample will ever provide less sampling error than a statistical sample of the same size.

Accepted Answer

Nonstatistical samples can sometimes provide less sampling error if they are carefully selected to be representative, though this is generally less reliable than statistical sampling methods.

Question 10

The population of incomes in a particular community is thought to be highly right-skewed with a mean equal to $36,789 and a standard deviation equal to $2,490. Based on this, if a sample of size n = 36 is selected, the highest sample mean that we would expect to see would be approximately $38,034.

Accepted Answer

The highest sample mean that we would expect to see is approximately $38,034, which is calculated using the formula mean + 2.58 * standard deviation / square root of sample size.

Question 11

A sampling distribution is the distribution of the individual values that are included in a sample from a population.

Accepted Answer

A sampling distribution is the distribution of a statistic (such as the mean, variance, etc.) calculated from samples of a given size drawn from a population, not the distribution of individual values within a sample.

Question 12

Although the concept of sampling distributions is an important concept in statistics, it is very unlikely that a decision maker will actually construct a sampling distribution in any practical business situation.

Accepted Answer

Sampling distributions are a theoretical concept used to understand the behavior of sample statistics. In practical business situations, decision makers often use statistical software or rely on specific sample data analyses rather than constructing sampling distributions manually.

Question 13

When a population is not normally distributed, the Central Limit Theorem states that a sufficiently large sample will result in the sample mean being normally distributed.

Accepted Answer

The Central Limit Theorem states that regardless of the original population distribution, the distribution of the sample means approaches a normal distribution as the sample size becomes sufficiently large.

Question 14

The population mean of income for adults in a particular community is known to be $28,600. Given this information, the sampling distribution of values will be less than this depending on the size of the sample used in developing the sampling distribution.

Accepted Answer

The sampling distribution of the mean will center around the population mean ($28,600), regardless of the sample size, due to the Central Limit Theorem. The variability of the sampling distribution will decrease with larger sample sizes, but the mean of the sampling distribution will not depend on the sample size.

Question 15

A major automobile manufacturer has developed a new model car that it claims will average 25 mpg on the highway. A random sample of fifty of these cars was tested and they averaged 24 mpg. This means that the claim made by the auto company is incorrect.

Accepted Answer

A single sample average of 24 mpg does not conclusively prove the company's claim of 25 mpg average is incorrect, as variability in sample results and measurement error could account for the difference.

Question 16

If a population standard deviation is 100, then the sampling distribution for will have a standard deviation that is less than 100 for all sample sizes greater than 2.

Accepted Answer

The standard deviation of the sampling distribution, also known as the standard error, decreases as the sample size increases, due to the formula for the standard error being the population standard deviation divided by the square root of the sample size (n). Thus, for any sample size greater than 2, the standard deviation of the sampling distribution will be less than the population standard deviation of 100.

Question 17

The population of incomes in a particular community is thought to be highly right-skewed with a mean equal to $36,789 and a standard deviation equal to $2,490. Based on this, if a sample of size n = 36 is selected, the sampling distribution would have a mean equal to the population mean, but the standard deviation of the sampling distribution will be one-sixth of the population standard deviation.

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean, and the standard deviation of the sampling distribution (standard error) is equal to the population standard deviation divided by the square root of the sample size, n. Here, sqrt(36) = 6, so the standard deviation of the sampling distribution is indeed one-sixth of the population standard deviation.

Question 18

A sampling distribution for a sample mean shows the distribution of the possible values for the sample mean for a given sample size from a population.

Accepted Answer

A sampling distribution of the sample mean is indeed a probability distribution that shows all the possible values of the mean of a specific sample size drawn from a population, illustrating how the sample mean varies from sample to sample.

Question 19

The sampling distribution for is actually the distribution of possible sampling error for samples of a given size selected at random from the population.

Accepted Answer

The sampling distribution represents the distribution of a statistic (like the sample mean) over many samples drawn from the same population, reflecting the variability or error in the statistic due to sampling.

Question 20

If a population is normally distributed, then the sampling distribution for the sample mean will always be normally distributed regardless of the sample size.

Accepted Answer

According to the Central Limit Theorem, if a population is normally distributed, then the sampling distribution of the sample mean will also be normally distributed, regardless of the sample size.

Quiz 7: Introduction to Sampling Distributions