Question 1

Which of the following best describes the Central Limit Theorem?
A)If the distribution of a random variable is non-normal, the sampling distribution of the sample mean will be approximately normal for samples n ≥ 30.
B)The sample mean is close to 0.50.
C)The underlying population is normal.
D)All of these choices are correct.

Accepted Answer

The Central Limit Theorem states that for a large enough sample size (n ≥ 30), the sampling distribution of the sample mean will be approximately normal, regardless of the underlying population distribution. This allows for the use of inferential statistical methods that assume normality, even when the original data may not be normally distributed. Option B is not a correct description of the theorem, and Option C is not always true as the theorem does not require the underlying population to be normal. Option D is not the correct answer as only Option A is correct.

Question 2

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean $\mu$:
A)when the population is normally distributed.
B)when the population is symmetric.
C)when the population size N > 30.
D)for all populations.

Accepted Answer

The expected value of the sampling distribution of the sample mean is always equal to the population mean, regardless of the population's distribution. This is a fundamental principle of sampling distributions.

Question 3

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:
A)160 and 32.
B)64 and 32.
C)160 and 4.
D)20 and 4.

Accepted Answer

The mean of the sample means is equal to the population mean, which is 160. The standard error of the sample mean is calculated as the population standard deviation divided by the square root of the sample size, which is 32 / sqrt(64) = 32 / 8 = 4.

Question 4

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:
A)12.649.
B)25.0.
C)2.56.
D)3.125.

Accepted Answer

The standard deviation of the sample mean (standard error) is calculated by dividing the population standard deviation by the square root of the sample size: 25 / √64 = 3.125.

Question 5

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true?
A)The population standard deviation decreases.
B)The standard error of the sample mean decreases.
C)The population standard deviation increases.
D)The standard error of the sample mean increases.

Accepted Answer

As n increases, the standard error of the sample mean decreases. This is because the formula for the standard error of the sample mean includes a denominator of the square root of n, so as n increases, the denominator gets larger and therefore the standard error gets smaller. The population standard deviation may or may not change as a result of the increase in sample size.

Question 6

Which of the following are factors that determine which statistical technique should be used?
A)The type of data being measured.
B)The purpose of the statistical inference.
C)The type of data being measured and the purpose of the statistical inference.
D)None of these choices are correct.

Accepted Answer

Both the type of data being measured and the purpose of the statistical inference are important factors in determining which statistical technique should be used. Different techniques are suited for different types of data, such as categorical vs. continuous, and different techniques are appropriate for different research questions, such as comparing means or testing for associations. Therefore, both factors should be considered in selecting an appropriate statistical method.

Question 7

If all possible samples of size n are drawn from an infinite population with mean $\mu$ and standard deviation $\sigma$, then the standard error of the sample mean is inversely proportional to: 
A)$\mu$.
B) $\sigma$
C) $n$.
D) $\sqrt { n }$

Accepted Answer

The standard error of the sample mean is calculated as the population standard deviation ($σ$) divided by the square root of the sample size ($n$), which is $\frac{σ}{\sqrt{n}}$. Therefore, it is inversely proportional to the square root of the sample size ($\sqrt{n}$).

Question 8

Which of the following are the two most important types of data?
A)Numerical data.
B)Numerical data and nominal data.
C)Nominal data.
D)None of these choices are correct.

Accepted Answer

Numerical data is important because it provides precise numerical measurements that can be analyzed mathematically. Nominal data is important because it allows for classification and categorization of data, which can help with organization and further analysis. Together, these two types of data provide a comprehensive understanding of the information being studied.

Question 9

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:
A)0.4332.
B)0.0668.
C)0.0987.
D)0.9013.

Accepted Answer

We can use the Central Limit Theorem to approximate the distribution of sample means as normal with a mean of 75 and a standard deviation of 12/√36=2. Thus, we want to find P(Z>(78-75)/2) = P(Z>1.5) ≈ 0.0668, where Z is a standard normal random variable.

Question 10

The finite population correction factor should not be used when:
A)we are sampling from an infinite population.
B)we are sampling from a finite population.
C)sample size is greater than 1% of the population size.
D)None of these choices are correct.

Accepted Answer

The finite population correction factor is used to adjust for the decrease in variability that occurs when sampling from a finite population. If we are sampling from an infinite population, there is no need for such adjustment, and therefore the correction factor should not be used.

Question 11

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:
A)9 and 45.
B)45 and 9.
C)81 and 45.
D)45 and 1.

Accepted Answer

The mean of the sampling distribution of the sample mean is equal to the population mean, which is 45. The standard error of the sampling distribution is calculated as the population standard deviation divided by the square root of the sample size, which is 9 / sqrt(81) = 9 / 9 = 1.

Question 12

The central limit theorem states that if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean $\bar { X }$ :
A)is approximately normal if n > 30.
B)is approximately normal if n < 30.
C)is approximately normal if the underlying population is normal.
D)has the same variance as the population.

Accepted Answer

The central limit theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large (commonly n > 30), regardless of the population's distribution.

Question 13

Which of the following best describes numerical statistical variables?
A)Variables whose outcomes are real numbers.
B)Variables which are nominal.
C)Variables which are categorical.
D)None of these choices are correct.

Accepted Answer

Numerical statistical variables are variables whose outcomes are numerical or real numbers. Examples include height, weight, age, and temperature.

Question 14

A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are 52 and 20, respectively. The probability that the sample mean will be larger than 49 is:
A)0.4452.
B)0.9452.
C)0.8554.
D)0.3554.

Accepted Answer

The answer of A sample of size 50 is to...

Question 15

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places):
A)13.00.
B)5.814.
C)2.600.
D)5.767.

Accepted Answer

The answer of A population that consists of 250 items...

Question 16

The standard deviation of the sampling distribution of the sample mean is also called the:
A)central limit theorem.
B)standard error of the mean.
C)finite population correction factor.
D)population standard deviation.

Accepted Answer

The standard deviation of the sampling distribution of the sample mean is also called the standard error of the mean. This measures the variability of sample means around the true population mean and gives an estimate of how much the sample mean deviates from the population mean.

Question 17

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:
A)0.1543.
B)0.0986.
C)0.9938.
D)0.0062.

Accepted Answer

The sample mean follows a normal distribution with mean 33 and standard deviation 6/sqrt(100) = 0.6. We calculate the z-scores for 34.5 and 36.1, then find the probability between these z-scores using the standard normal distribution table. The probability is very low, corresponding to option D.

Question 18

Which of the following are involved in statistical inference?
A)How to calculate relevant statistics.
B)When to apply a statistical technique.
C)Why statistical methods are used.
D)All of these choices are correct.

Accepted Answer

All of the choices are involved in statistical inference. Statistical inference involves understanding how to calculate relevant statistics, as well as when and why to apply various statistical techniques.

Question 19

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:
A)sample size is smaller than 10% of the population size.
B)population size is smaller than 10% of the sample size.
C)sample size is greater than 10% of the population size.
D)population size is greater than 10% of the sample size.

Accepted Answer

The finite population correction factor is applied when the sample size is greater than 10% of the population size, to adjust for the decrease in variability that occurs when sampling without replacement from a finite population.

Question 20

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is:
A)0.0166.
B)0.0228.
C)0.3708.
D)0.0394.

Accepted Answer

The probability can be found using the Z-score formula for a sample mean: Z = (X - μ) / (σ/√n), where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Here, Z = (56 - 60) / (10/√25) = -2. The probability of Z being less than -2, according to the Z-table, is approximately 0.0228. However, due to the choices provided and the context of the question, it seems there was a misunderstanding in the calculation or interpretation. The correct approach leads to a Z-score calculation, and the closest match to the Z-table probability for a Z-score of -2 is 0.0228, but the correct choice based on the options provided is 0.0166, indicating a possible error in the interpretation of the question or the options given.

Quiz 9: Statistical Inference and Sampling Distributions

Which of the following best describes the Central Limit Theorem?

The expected value of the sampling distribution of the sample mean $\bar { X }$ equals the population mean $\mu$ :

Random samples of size 64 are taken from an infinite population whose mean is 160 and standard deviation is 32. The mean and standard error of the sample mean, respectively, are:

Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample of size 64 is taken from this population. The standard deviation of the sample mean equals:

A sample of size n is selected at random from an infinite population. As n increases, which of the following statements is true?

Which of the following are factors that determine which statistical technique should be used?

If all possible samples of size n are drawn from an infinite population with mean $\mu$ and standard deviation $\sigma$ , then the standard error of the sample mean is inversely proportional to:

Which of the following are the two most important types of data?

Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:

The finite population correction factor should not be used when:

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample mean are:

The central limit theorem states that if a random sample of size n is drawn from a population, then the sampling distribution of the sample mean $\bar { X }$ :

Which of the following best describes numerical statistical variables?

A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are 52 and 20, respectively. The probability that the sample mean will be larger than 49 is:

A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of size 5 is taken at random from this population. The standard error of the sample mean equals (up to three decimal places) :

The standard deviation of the sampling distribution of the sample mean is also called the:

An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is to be taken at random from this population. The probability that the sample mean will be between 34.5 and 36.1 is:

Which of the following are involved in statistical inference?

As a general rule in computing the standard error of the sample mean, the finite population correction factor is used only if the:

A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10. The probability that the mean of a sample of 25 elements taken from this population will be smaller than 56 is: