Question 1

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

To find the probability of selecting a sample mean of 13 or larger, we use the formula for the standard error of the mean (SEM = σ/√n), where σ is the population standard deviation and n is the sample size. Here, SEM = 3.5/√49 = 0.5. The Z-score for a sample mean of 13 is (13 - 12)/0.5 = 2. A Z-score of 2 corresponds to a probability of less than .10 in the upper tail of the normal distribution, indicating that the probability of selecting a sample mean of 13 or larger is less than .10.

Question 2

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

Accepted Answer

This statement is true. A sample mean is calculated from a subset of the population and may not accurately represent the whole population, therefore making it unlikely to be equal to the population mean.

Question 3

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

Accepted Answer

This is the definition of a sampling distribution. It is the distribution of sample statistics (such as means or variances) from repeated random samples taken from the same population.

Question 4

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

We can use the Central Limit Theorem to determine the range of sample means:

mean of sampling distribution of x̄ = population mean = 38 
standard deviation of sampling distribution of x̄ = population standard deviation / sqrt(sample size) = 20 / sqrt(100) = 2

68% of sample means will fall within one standard deviation of the mean:

38 - 2 = 36 
38 + 2 = 40

Therefore, the range of sample means in which 68% of the sample means should fall is between 36 and 40. Thus, choice D is the correct answer.

Question 5

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 non-probability sampling, specifically the non-probability sampling technique known as convenience sampling. In convenience sampling, the order in which participants are selected doesn't matter, and the same individual participants can be included multiple times. However, this sampling technique does not allow for generalization of results beyond the sample that was selected.

Question 6

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 probability of selecting a sample mean that is at least equal to the population mean is .50 because, by the Central Limit Theorem, the sampling distribution of the sample mean is normally distributed if the sample size is large enough (which is generally considered to be the case for n > 30). This means the distribution is symmetric around the population mean, making the probability of selecting a sample mean at least equal to the population mean the same as the probability of selecting a sample mean at most equal to the population mean, both being .50.

Question 7

An article 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 value reported as "SEM" in the statement is 2.3, which stands for the standard error of the mean. The SEM is a measure of how much the sample mean deviates from the true population mean, and it is typically smaller than the sample standard deviation when the sample size is large enough. Therefore, the correct answer is C.

Question 8

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 9

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 10

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 number 40 is identified as the M SEM (Mean Standard Error), which represents the standard deviation of the sample means. Therefore, option B is the correct answer.

Question 11

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 12

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

Accepted Answer

Theoretical sampling does not determine the number of samples; it's a qualitative research method focusing on selecting cases that help the researcher develop a theory. In quantitative terms, from a population of 2 people, the number of possible samples of size 5 (assuming replacement) would be determined by permutations or combinations depending on the context, but not related to theoretical sampling principles.

Question 13

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

Accepted Answer

The standard error of the mean is typically reported in a graph using error bars. This helps to visually represent the variability of the data, and how accurate the sample mean represents the population mean.

Question 14

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

The statement is true. The sampling distribution of the mean is a theoretical probability distribution that represents the distribution of all the possible means that could be computed from an infinite number of samples of a given size drawn from the same population. It is an important concept in inferential statistics because it allows us to make probabilistic statements about the population based on the characteristics of the sample mean.

Question 15

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 the selection of a wider range of samples as it does not rely on specific pre-determined conditions or treatments, whereas experimental sampling is limited to the samples that can be tested under the specific experimental conditions. Therefore, as the sample size increases, the number of possible samples that can be selected through theoretical sampling also increases, while experimental sampling is still limited by the same conditions, resulting in fewer possible samples.

Question 16

A researcher selects a sample of nine 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

According to the Central Limit Theorem, the sampling distribution of the means will be approximately normal with a mean of 8 and a standard error of σ/√n = 3/√9 = 1. Therefore, we can use the z-score formula to find the proportion of sample means between two values.

For choice C: (10-8)/1 = 2, which corresponds to a probability of 0.4772. To find the probability between 9 and 10, we subtract the probability of less than 9 from the probability of less than 10: 0.4772 - 0.1587 = 0.3185 or about 31.85%. Therefore, about 14% of the sample means will fall between 9 and 10, which satisfies the condition.

For other choices, we can use the same method to find the probabilities:
A) (6-8)/1 = -2, P(z < -2) = 0.0228
B) (5-8)/1 = -3, P(z < -3) = 0.0013
D) (11-8)/1 = 3, P(z < 3) = 0.9987

As we can see, only choice C satisfies the condition of having about 14% of sample means between the given range.

Question 17

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 18

The sampling distribution of the mean has minimum variance.

Accepted Answer

The sampling distribution of the mean has a minimum variance as it is less variable than the original population distribution. This is because the sample mean is less affected by chance fluctuations in individual observations than the individual observations themselves. Therefore, the more samples we take, the less variable the means of those samples will be, leading to a smaller variance in the sampling distribution of the mean.

Question 19

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 z transformation formula is used to find the probability of a sample mean falling within a certain range. The formula is:$ z=\frac{M-\mu}{\sigma_{M}} $where:* $M$ is the sample mean* $\mu$ is the population mean* $\sigma_{M}$ is the standard deviation of the sampling distribution of sample meansAll of the answer choices provided are correct notations for the z transformation formula.

Question 20

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

Accepted Answer

A sampling distribution is a distribution of statistics (e.g. means or proportions) calculated from repeated samples of the same size from a population. It is not a distribution of scores in a population.

Quiz 7: Probability and Sampling Distributions