Question 1

When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample means will be normal, regardless of sample size n.

Accepted Answer

This statement is true according to the Central Limit Theorem.

Question 2

The standard deviation of a sampled population is also called the standard error of the sample mean.

Accepted Answer

The standard deviation of a sampled population is different from the standard error of the sample mean. The standard error of the sample mean is the standard deviation of the sampling distribution of the sample mean, which is calculated by dividing the standard deviation of the population by the square root of the sample size.

Question 3

If all possible samples of size n are drawn from an infinite population with a mean of 60 and a standard deviation of 8, then the standard error of the sample mean equals 1.0 only for samples of size 64.

Accepted Answer

The formula for standard error of the sample mean is standard deviation / square root of sample size.

Therefore, for a population with standard deviation of 8:

- If the sample size is n = 64, the standard error is 8 / sqrt(64) = 1. 
- For any other sample size, the standard error will be lower or higher than 1.

Hence, the statement is true.

Question 4

The standard error of the mean is the standard deviation of the sampling distribution of $\bar { X }$ .

Accepted Answer

The standard error of the mean measures how much the sample mean ($\bar { X }$) is expected to vary from the true population mean. It is indeed the standard deviation of the sampling distribution of the sample mean ($\bar { X }$).

Question 5

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is called the:
A)standard error of $\bar { X }$ .
B)expected value of $\bar { X }$ .
C)sampling distribution of $\bar { X }$ .
D)normal distribution.

Accepted Answer

The sampling distribution of $\bar{X}$ refers to the probability distribution of all possible sample means taken from a population. This concept is fundamental in inferential statistics, allowing for the estimation of population parameters based on sample statistics.

Question 6

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean $\bar { X }$ will be:
A)normal for all values of n.
B)normal only for n > 30.
C)approximately normal for all values of n.
D)approximately normal only for n > 30.

Accepted Answer

The sampling distribution of the sample mean $\bar { X }$ is normal for all values of n when the population from which the sample is drawn is normal, according to the Central Limit Theorem. This holds true regardless of the sample size n.

Question 7

The sampling distribution of the sample proportion is always normally distributed.

Accepted Answer

The sampling distribution of the sample proportion follows a normal approximation when the sample size is large enough (n≥30) and the population proportion is not too close to 0 or 1. However, it is not always exactly normally distributed.

Question 8

Which of the following best describes the sampling distribution of the sample proportion?
A)The sampling distribution of the sample proportion is always normally distributed.
B)The sampling distribution of the sample proportion is approximately normally distributed provided that np≥ 5 and nq≥ 5.
C)The sampling distribution of the sample proportion is skewed
D)None of these choices are correct.

Accepted Answer

The distribution of the sample proportion will be approximately normally distributed if the sample size is large enough and both np and nq are greater than or equal to 5. This is known as the npq rule. If the sample size is too small or npq<5, the sampling distribution may be skewed. Therefore, option B is the best choice.

Question 9

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?
A)14.28%
B)0.59%
C)100%
D)None of these choices are correct

Accepted Answer

This problem can be solved using binomial distribution with n=35 and p=0.05. We are looking for the probability of more than 5 defective products, which can be calculated as 1 minus the cumulative probability of 5 or fewer defective products. Using a calculator or software, the probability is found to be 0.59%, which is choice B.

Question 10

A sample of size 70 is selected at random from a finite population. If the finite population correction factor is 0.808, the population size (rounded to the nearest integer) must be 200.

Accepted Answer

The finite population correction factor (FPC) is calculated as $\sqrt{\frac{N-n}{N-1}}$, where $N$ is the population size and $n$ is the sample size. Given an FPC of 0.808 and a sample size of 70, solving for $N$ does not yield exactly 200.

Question 11

The how, when and why of statistical inference puts the problem into context.

Accepted Answer

Statistical inference involves making conclusions or predictions about a population based on a sample. Understanding the how, when, and why of statistical inference is crucial for making informed decisions and drawing accurate conclusions. It helps to put the problem into context and ensures that the chosen statistical analysis is appropriate for the data and research question.

Question 12

A sample of size n is selected at random from an infinite population. As n increases the standard error of the sample mean decreases.

Accepted Answer

As the sample size increases, the standard error of the sample mean decreases because the sample mean becomes more representative of the population mean. This is due to the Central Limit Theorem, which states that as sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

Question 13

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean $\bar { X }$ is referred to as the normal distribution.

Accepted Answer

The distribution of the sample mean $\bar{X}$ approaches a normal distribution as the sample size n increases, according to the Central Limit Theorem, regardless of the population's distribution, but it is not automatically normal for all possible samples unless the population itself is normally distributed.

Question 14

It is known that 40% of voters in a certain electorate are in favour of a particular candidate. If a sample of size 30 is taken, what is the probability that less than 35% are in favour of this political candidate?
A)40%
B)Less than 0.10%
C)28.77%
D)None of these choices are correct.

Accepted Answer

This is a binomial probability problem that can be approximated using the normal distribution because the sample size is large enough. The mean of the distribution ($\mu$) is $np = 30 \times 0.4 = 12$ and the standard deviation ($\sigma$) is $\sqrt{np(1-p)} = \sqrt{30 \times 0.4 \times 0.6} \approx 2.683$. To find the probability of less than 35% (which is 10.5 out of 30), we convert this to a z-score and look up the probability in a standard normal distribution table or use a calculator. The calculation of the z-score is $\frac{10.5 - 12}{2.683}$, which gives a value that corresponds to a probability of approximately 28.77%. This is a simplified explanation; the exact calculation involves more detailed steps with the normal distribution.

Question 15

The type of data being measured is one of the first key factors of determining the statistical methods that should be used.

Accepted Answer

The type of data being measured (categorical or numerical) is a crucial factor in deciding which statistics to use. For example, categorical data may call for proportions and percentages, while numerical data may require measures of central tendency and variability.

Question 16

The purpose behind the statistical technique is irrelevant in statistical inference.

Accepted Answer

The purpose behind the statistical technique is crucial in statistical inference as it guides the choice of methods and interpretations of results to address specific research questions or hypotheses.

Question 17

The central limit theorem is basic to the concept of statistical inference, because it permits us to draw conclusions about the population based strictly on sample data, and without having any knowledge about the distribution of the underlying population.

Accepted Answer

The central limit theorem provides a way to make statistical inferences about a population based on sample data, even when we don't know the distribution of the population. Therefore, the statement is true.

Question 18

If it is known that the population proportion of car parks in a particular city that offer early discount fees is 25%, which of the following best describes the mean and standard deviation of the sampling distribution of the sample proportion of car parks that offer early discount fees for samples taken of size 30?
A)The mean is 0.25 and the standard deviation is 0.006.
B)The mean is 0.25 and the standard deviation is 0.75.
C)The mean is 0.25 and the standard deviation is 0.079.
D)None of these choices are correct.

Accepted Answer

The mean of the sampling distribution of sample proportions is equal to the population proportion, which is 0.25 in this case. The standard deviation of the sampling distribution of the sample proportion can be calculated using the formula:

standard deviation = sqrt(p(1-p)/n)

where p is the population proportion and n is the sample size. Plugging in the values, we get:

standard deviation = sqrt(0.25(1-0.25)/30) = 0.079

Therefore, the best choice is C: The mean is 0.25 and the standard deviation is 0.079.

Question 19

The objective of statistical inference is to use a sample statistic to draw conclusions about a population parameter of interest.

Accepted Answer

This is a accurate statement describing the purpose of statistical inference.

Question 20

Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample of size 50 is taken from this population. The standard deviation of the sample mean equals 3.2.

Accepted Answer

The answer of Consider an infinite population with a mean...

Quiz 9: Statistical Inference and Sampling Distributions

When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample means will be normal, regardless of sample size n.

The standard deviation of a sampled population is also called the standard error of the sample mean.

If all possible samples of size n are drawn from an infinite population with a mean of 60 and a standard deviation of 8, then the standard error of the sample mean equals 1.0 only for samples of size 64.

The standard error of the mean is the standard deviation of the sampling distribution of Xˉ\bar { X }Xˉ .

If all possible samples of size n are drawn from a population, the probability distribution of the sample mean Xˉ\bar { X }Xˉ is called the:

If a random sample of size n is drawn from a normal population, then the sampling distribution of the sample mean Xˉ\bar { X }Xˉ will be:

The sampling distribution of the sample proportion is always normally distributed.

Which of the following best describes the sampling distribution of the sample proportion?

A manufacturing company is concerned about the number of defective items produced by their assembly line. In the past they have had 5% of their products produced defectively. They take a random sample of 35 products. What is the probability that more than 5 products in the sample are defective?