Question 1

There are 25 possible samples of size 2 that can be selected from a population of 5 people using theoretical sampling.

Accepted Answer

Theoretical sampling does not involve actually selecting samples from the population, but rather using statistical methods to infer information about the population from the available data. Therefore, the concept of "possible samples of size 2" does not apply to theoretical sampling.

Question 2

A researcher selects a sample of 100 participants from a population with a mean of 38 and a standard deviation of 20.About 68% of the sample means in this sampling distribution should be between a sample mean of

Accepted Answer

The answer of A researcher selects a sample of 100...

Question 3

A researcher selects a sample of 9 participants from a population with a mean of 8 and a standard deviation of 3.About 14% of the sample means in this sampling distribution should be between a sample mean of

Accepted Answer

The answer of A researcher selects a sample of 9...

Question 4

A researcher selects a sample of 49 participants from a population with a mean of 12 and a standard deviation of 3.5.What is the probability of selecting a sample mean that is at least equal to the population mean?

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be normally distributed with the same mean as the population mean. Therefore, the probability of selecting a sample mean that is at least equal to the population mean is 50%, which is equal to the probability of selecting a sample mean that is at most equal to the population mean.

Question 5

An articles states that a sample of 40 participants took 12 ± 2.3 (M ± SEM)seconds to complete a cognitive assessment.Which value is the standard error of the mean?

Accepted Answer

The SEM (standard error of the mean) is reported as 2.3 in the article.

Question 6

For the experimental sampling strategy,the order in which a participant is selected does not matter,and once selected,each participant is not replaced before sampling again.

Accepted Answer

This is the definition of a non-probability sampling technique called purposive sampling or judgmental sampling. The order in which participants are selected does not affect the sampling process, and once selected, they are not replaced before the next sampling.

Question 7

A sampling distribution is a distribution of scores in a population.

Accepted Answer

A sampling distribution is not a distribution of scores in a population, but rather a distribution of sample statistics (such as means or proportions) that would be obtained if an infinite number of random samples were drawn from the same population.

Question 8

A researcher selects a sample of 25 participants from a population with a mean of 10 and a standard deviation of 5.What is the probability of selecting a sample mean that is at least two standard deviations larger than the population mean?

Accepted Answer

The probability of selecting a sample mean that is at least two standard deviations larger than the population mean is found using the standard normal distribution (Z-distribution). The Z-score for two standard deviations above the mean is 2. The Central Limit Theorem allows us to treat the distribution of sample means as normally distributed if the sample size is sufficiently large (n ≥ 30 is a common rule of thumb, but n = 25 is often considered large enough for the approximation to be useful). The standard error of the mean (SEM) is the standard deviation of the population divided by the square root of the sample size. In this case, the SEM = 5 / sqrt(25) = 1. The Z-score for a sample mean that is two standard deviations (10 units in this context, since the population standard deviation is 5 and we're considering 2 of these) above the population mean of 10 is (20 - 10) / 1 = 10. However, since we're looking for the probability of being at least two standard deviations above, we're interested in the tail beyond Z = 2. Looking up the Z-score of 2 in standard normal distribution tables or using a calculator gives us the area to the left of Z = 2, which we subtract from 1 to find the tail probability. This gives us approximately 0.0228, indicating a small probability of selecting such a sample mean by chance.

Question 9

The central limit theorem states that the sample mean will equal the population mean on average.

Accepted Answer

The answer of The central limit theorem states that the...

Question 10

A researcher selects a sample of 49 participants from a population with a mean of 12 and a standard deviation of 3.5.What is the probability of selecting a sample mean of 13 or larger from this population?

Accepted Answer

We can use the formula for the standard error of the mean, which is σ/√n, where σ is the population standard deviation and n is the sample size. Plugging in the values, we get a standard error of the mean of 0.5. To find the probability of selecting a sample mean of 13 or larger, we need to standardize this value by subtracting the population mean from 13 and dividing by the standard error of the mean: (13-12)/0.5 = 2. The probability of selecting a sample mean of 13 or larger is then equal to the probability of selecting a value 2 standard deviations or more above the mean in a normal distribution. This probability is less than 0.03 (found using a z-table or calculator). Therefore, the answer is C.

Question 11

Because the sampling distribution of sample means is normally distributed,we can use the z transformation formula to find probabilities.Which of the following gives the appropriate notation for this formula?

Accepted Answer

The answer of Because the sampling distribution of sample means...

Question 12

The sampling distribution of the mean has minimum variance.

Accepted Answer

The sampling distribution of the mean has the minimum variance compared to other sampling distributions, such as the sampling distribution of the individual observations. This is due to the central limit theorem, which states that as sample size increases, the sampling distribution of the mean approaches a normal distribution with a mean equal to the population mean and a variance equal to the population variance divided by the sample size. Therefore, the variance of the sampling distribution of the mean decreases as sample size increases.

Question 13

A sample mean is not always equal to a population mean.

Accepted Answer

This is true. A sample mean is the average value of a subset of data, while the population mean is the average value of the entire population. It is possible for the sample mean to be close to the population mean, but it is not always equal.

Question 14

A researcher selects a sample of 25 participants from a population with a mean of 10 and a standard deviation of 5.What is the probability of selecting a sample mean equal to at least 12?

Accepted Answer

The answer of A researcher selects a sample of 25...

Question 15

A researcher states that a sample of 100 shoppers spend 112 ± 40 (M ± SEM)dollars on groceries each week.Which value is the standard error of the mean?

Accepted Answer

The standard error of the mean is given as ±40 in the statement. Therefore, the correct answer is B, which indicates that the standard error of the mean is reported as 40 in the article.

Question 16

There are more possible samples that can be selected using theoretical sampling than using experimental sampling for any sample size greater than 1.

Accepted Answer

Theoretical sampling allows for a greater variety and flexibility in selecting samples as it involves selecting samples based on emerging theories and concepts rather than pre-determined variables. Experimental sampling is more limited in its options for sample selection as it is based on pre-determined variables and conditions. Therefore, for any sample size greater than 1, theoretical sampling allows for more possible samples to be selected.

Question 17

A sampling distribution is a distribution of all sample means or sample variances that could be obtained in samples of a given size from the same population.

Accepted Answer

A sampling distribution is a distribution of all possible sample means or sample variances that could be obtained in samples of a given size from the same population. This distribution shows how the sample statistics vary from sample to sample, and it is important for making inferences about a population based on a sample.

Question 18

The sampling distribution of the mean is a distribution of all the possible samples of a particular size that can be selected from a given population.

Accepted Answer

This statement accurately describes the concept of the sampling distribution of the mean.

Question 19

Researchers typically sample with replacement,although the development of statistical theory was based on the use of sampling without replacement.

Accepted Answer

Sampling with replacement is not typically used in most research contexts; instead, sampling without replacement is more common because it better reflects the finite nature of most populations under study. Statistical theory, including formulas for variances and standard errors, often accounts for both sampling methods, but practical research, especially involving finite populations, frequently relies on sampling without replacement to ensure all individuals or elements have a chance to be selected without being chosen multiple times.

Question 20

How is the standard error of the mean typically reported in a graph?

Accepted Answer

The standard error of the mean is commonly reported using error bars in a graph. This allows the viewer to quickly see the variability around the sample mean and helps to determine if the mean is statistically significant or not.

Quiz 7: Probability and Sampling Distributions