Question 1

If p = .8 and n = 50,then we can conclude that the sampling distribution of 
is approximately a normal distribution.

Accepted Answer

According to the central limit theorem, when n is sufficiently large (usually considered to be greater than or equal to 30) and p is not too close to 0 or 1 (usually considered to be between 0.1 and 0.9), the sampling distribution of a proportion is approximately normal. In this case, n = 50 is sufficiently large and p = 0.8 is not too close to 0 or 1, so we can conclude that the sampling distribution of the proportion is approximately normal.

Question 2

A sample statistic is an unbiased point estimate of a population parameter if the mean of the populations of all possible values of the sample statistic equals the population parameter.

Accepted Answer

This is the definition of an unbiased estimator. A sample statistic is an unbiased estimator if the expected value of the sample statistic equals the population parameter.

Question 3

A minimum-variance unbiased point estimate has a variance that is as small as or smaller than the variances of any other unbiased point estimate.

Accepted Answer

This is the definition of a minimum-variance unbiased point estimate. If it did not have a smaller variance than any other unbiased point estimate, it would not be considered minimum-variance.

Question 4

The sampling distribution of 
must be a normal distribution with mean = 0 and standard deviation = 1.

Accepted Answer

The sampling distribution of a statistic depends on the distribution of the population and the sample size. It is not always a standard normal distribution with mean 0 and standard deviation 1.

Question 5

If the sampled population is exactly normally distributed,then the sampling distribution of 
is also expected to be normal,regardless of the sample size.

Accepted Answer

If the population is normally distributed, then the sampling distribution will also be normal regardless of the sample size due to the central limit theorem.

Question 6

The Central Limit Theorem states that as sample size increases,the population distribution more closely approximates a normal distribution.

Accepted Answer

The Central Limit Theorem states that as sample size increases, the sampling distribution of the mean or sum will approach a normal distribution regardless of the shape of the population distribution. This is because the mean or sum of a large sample will have less variability and be less influenced by extreme values, leading to a more normal distribution.

Question 7

If p = .9 and n = 40,then we can conclude that the sampling distribution of 
is approximately a normal distribution.

Accepted Answer

The answer of If p = .9 and n =...

Question 8

If the sampled population distribution is skewed,then in most cases the sampling distribution of the mean can be approximated by the normal distribution if the sample size n is at least 30.

Accepted Answer

This is known as the Central Limit Theorem, which states that for large sample sizes (generally n >= 30), the sampling distribution of the mean will be approximately normal regardless of the shape of the population distribution.

Question 9

The standard deviation of all possible sample proportions increases as the sample size increases.

Accepted Answer

The standard deviation of all possible sample proportions decreases as the sample size increases.

Question 10

For any sampled population,the population of all sample means is approximately normally distributed.

Accepted Answer

Only for populations that are a normal distribution. According to the CLT, if the sample sizes are 30 or greater, the sample means are approximately normally distributed.

Question 11

The sampling distribution of the sample mean is developed by repeatedly taking samples of size n and computing the sample means and reporting the resulting sample means in the form of a probability distribution.

Accepted Answer

This statement accurately describes the process for developing the sampling distribution of the sample mean.

Question 12

If we have a sample size of 100 and the estimate of the population proportion is .10,we can estimate the sampling distribution of 
with a normal distribution.

Accepted Answer

If the sample size is large (n > 30) and the estimate of the population proportion is not too close to 0 or 1, then we can use the normal distribution to estimate the sampling distribution. In this case, the sample size is 100 which exceeds 30 and the estimate of the population proportion is not too close to 0 or 1 (.10 is not too far from .50), so a normal distribution can be used to estimate the sampling distribution.

Question 13

If a population is known to be normally distributed,then it follows that the sample standard deviation must equal σ.

Accepted Answer

While the population standard deviation is denoted by σ, the sample standard deviation is denoted by s. They are not necessarily equal, although as the sample size approaches infinity, s will approach σ.

Question 14

As the sample size increases,the standard deviation of the sampling distribution increases.

Accepted Answer

As the sample size increases, the standard deviation of the sampling distribution actually decreases. This is known as the Central Limit Theorem. The sampling distribution becomes more normally distributed, and the spread of the distribution decreases as the sample size increases.

Question 15

The standard deviation of the sampling distribution of the sample mean is σ.

Accepted Answer

The standard deviation of the sampling distribution of the sample mean is σ/.

Question 16

A sample size of 500 is sufficiently large to conclude that the sampling distribution of 
is a normal distribution,when the estimate of the population proportion is .995.

Accepted Answer

A sample size of 500 is not sufficiently large when the population proportion is extremely close to 0 or 1, such as 0.995, because the normal approximation may not be accurate in such cases.

Question 17

The sampling distribution of a sample statistic is the probability distribution of the population of all possible values of the sample statistic.

Accepted Answer

The sampling distribution of a sample statistic represents the probabilities of all possible values of the sample statistic, based on repeated random sampling from the same population. This distribution can be used to make inferences about the population parameter that the sample statistic represents.

Question 18

The reason sample variance has a divisor of n − 1 rather than n is that it makes the variance an unbiased estimate of the population variance.

Accepted Answer

This is true – using n - 1 as the divisor for the sample variance adjusts for the fact that the sample mean is also being estimated, resulting in a more accurate estimate of the population variance.

Question 19

The mean of the sampling distribution of 
is always equal to the mean of the sampled population.

Accepted Answer

This is a fundamental property of the sampling distribution of the mean, which is a distribution of means obtained from multiple samples drawn from the same population. The central limit theorem states that as the sample size increases, the distribution of sample means will approach a normal distribution with a mean equal to the population mean. Therefore, the mean of the sampling distribution will also be equal to the population mean.

Question 20

If a population is known to be normally distributed,then it follows that the sample mean must equal the population mean.

Accepted Answer

The sample mean is an estimate of the population mean and may not be exactly equal due to sampling variability.

Quiz 7: Sampling and Sampling Distributions

If p = .8 and n = 50,then we can conclude that the sampling distribution of is approximately a normal distribution.

A sample statistic is an unbiased point estimate of a population parameter if the mean of the populations of all possible values of the sample statistic equals the population parameter.

A minimum-variance unbiased point estimate has a variance that is as small as or smaller than the variances of any other unbiased point estimate.

The sampling distribution of must be a normal distribution with mean = 0 and standard deviation = 1.

If the sampled population is exactly normally distributed,then the sampling distribution of is also expected to be normal,regardless of the sample size.