Question 1

Which of the following is another name for assignable cause variation?

Accepted Answer

Assignable cause variation is also known as special cause variation. This type of variation is caused by factors that are not inherent to the process, but instead come from outside sources such as a broken machine or human error. It is distinct from common cause variation, which is a result of factors that are inherent to the process and is typically characterized by a stable, predictable range of outcomes. Controlled variation refers to the ability to manage and minimize variation within a process. Random variation is another term for common cause variation. Therefore, the best answer is B, special cause variation.

Question 2

The spread of the distribution of sample means is considerably less than the spread of the sampled population.

Accepted Answer

This statement is true, as the Standard Error of the Mean (SEM) causes the distribution of sample means to become more narrow and centered around the population mean.

Question 3

If random samples of size n = 36 are drawn from a non-normal population with finite mean  = 75 and standard deviation  = 15, then the sampling distribution of the sample mean  is approximately normally distributed with mean  = 75 and standard deviation  = 2.5.

Accepted Answer

This is a well-known result of the central limit theorem, which states that for large enough sample sizes, the sampling distribution of the sample mean will be approximately normal regardless of the underlying population distribution. The mean and standard deviation of this sampling distribution can be calculated using formulas and are given in the question.

Question 4

For a  control chart, where are the lower and upper control limits usually set?

Accepted Answer

The lower and upper control limits on a 11eab386_b93c_a44a_aec0_6fd64653c83e_TB4962_11 control chart are usually set at three standard deviations from the centerline. This allows for 99.7% of the data to fall within the control limits if the process is in control.

Question 5

According to the Central Limit Theorem, for large samples, the standard error of the sample mean is the population standard deviation divided by the square root of the sample size.

Accepted Answer

This statement is true. The Central Limit Theorem states that for large samples, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution, and the standard error of the sample mean is calculated by dividing the population standard deviation by the square root of the sample size.

Question 6

If the standard error of the sampling distribution of the sample proportion is 0.02049 for samples of size 500, then what may we conclude about the population proportion?

Accepted Answer

The standard error of the sample proportion is calculated as sqrt[(p(1-p))/n]. Given the standard error is 0.02049 and n is 500, solving for p gives approximately 0.3 or 0.7.

Question 7

What are some of the most common sources of random variation?

Accepted Answer

Random variation can come from both people and materials. For example, if a group of people are asked to complete a task, each person's approach or skill level may vary, leading to different results. Similarly, if different batches of a material are used in an experiment, there may be variations in the properties or composition of each batch, leading to different outcomes.

Question 8

As a general rule, the normal distribution provides a good approximation to the sampling distribution of the sample proportion,  , only if which of the following conditions holds?

Accepted Answer

The normal distribution approximation for the sampling distribution of the sample proportion holds only if np and n(1 - p) are both greater than 5, regardless of the sample size, population proportion or standard deviation of the population. This is known as the "success-failure" condition, where np and n(1 - p) represent the expected number of successes and failures in the sample, respectively. When these numbers are small, the distribution of the sample proportion becomes skewed and the normal approximation is no longer valid.

Question 9

Given a population proportion of p = 0.8 and a sample size of n = 100, what is the standard deviation of the sampling distribution of the sample proportion,  ?

Accepted Answer

The answer of Given a population proportion of p =...

Question 10

The Central Limit Theorem does not apply to the sample means of large samples drawn from a discrete distribution.

Accepted Answer

The Central Limit Theorem applies to the sample means of large samples drawn from any distribution, including discrete distributions.

Question 11

Twenty-five samples of size 1000 each were drawn from a manufacturing process and the number of defectives in each sample was counted. The average sample proportion was 0.05. What would be the upper control limit for the p chart?

Accepted Answer

The upper control limit (UCL) for a p-chart is calculated using the formula: UCL = p̄ + z * sqrt((p̄(1-p̄))/n), where p̄ is the average sample proportion, z is the z-value for the desired confidence level (typically 3 for 99.73% confidence), and n is the sample size. Here, p̄ = 0.05, n = 1000, and z = 3. Calculating gives UCL = 0.05 + 3 * sqrt((0.05 * 0.95) / 1000) = 0.0707.

Question 12

The most important contribution of the Central Limit Theorem is in statistical inference.

Accepted Answer

The Central Limit Theorem is an important tool in statistical inference as it allows us to make conclusions about a population based on a sample. It states that as the sample size increases, the sample mean approaches a normal distribution, regardless of the population distribution. This enables us to use common statistical methods that assume normality, such as hypothesis testing and confidence intervals, even when the population distribution is unknown or non-normal.

Question 13

If random samples of size n = 50 are drawn from a non-normal population with finite mean  = 100 and standard deviation  = 20, then the sampling distribution of the sum of sample measurements  is approximately normally distributed with mean  = 5000 and standard deviation  = 1000.

Accepted Answer

The answer of If random samples of size n =...

Question 14

In a recent study, it was reported that the proportion of employees who miss work on Fridays is 0.15, and that the standard deviation of the sampling distribution of sample proportion,  , is 0.025. What was the sample size, n?

Accepted Answer

The answer of In a recent study, it was reported...

Question 15

Sixty samples of size 600 each were drawn from a manufacturing process and the number of defectives in each sample was counted. The average sample proportion was 0.04. What would be the lower control limit for the p chart?

Accepted Answer

To find the lower control limit for the p chart, we use the formula: LCL= p̅-3√(p̅(1-p̅)/n)
where p̅ is the average sample proportion and n is the sample size.
LCL=0.04-3√(0.04*0.96/600)
LCL=0.04-0.016
LCL=0.024
Therefore, the lower control limit for the p chart is 0.024, which is closest to option B (0.016).

Question 16

The sampling distribution of the sample mean is exactly normally distributed, regardless of the sample size n.

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases, according to the Central Limit Theorem. For small sample sizes, the distribution of the sample mean will more closely resemble the population distribution, which may not be normal.

Question 17

The mean of the sample means and the standard deviation of 50 samples of size 5 taken from a production process under control are found to be 300 and 25, respectively. At which of the following values would the lower control limit for the  chart be located?

Accepted Answer

The answer of The mean of the sample means and...

Question 18

A statistics professor has stated that 90% of his students pass the class. To check this claim, a random sample of 150 students indicated that 129 passed the class. If the professor's claim is correct, what is the probability that 129 or fewer will pass the class this semester?

Accepted Answer

The probability can be calculated using the normal approximation to the binomial distribution. With a sample size of 150 and a success probability of 0.9, the mean is 135 and the standard deviation is approximately 3.674. Using the normal distribution, the probability of 129 or fewer passing is approximately 0.0516.

Question 19

Fifty samples of size 500 were drawn from a manufacturing process and the number of defectives in each sample was counted. The average of the sample proportion was 0.032. At which of the following values would the centreline for the p chart be located?

Accepted Answer

The centerline for a p chart is calculated as the average of the sample proportions, which is given as 0.032. Therefore, the centerline would be located at 0.032. The other options are not relevant in this case.

Question 20

The Central Limit Theorem describes the distribution of the sample mean except for populations that are normal.

Accepted Answer

The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population, as long as the sample size is large enough (usually at least 30).

Quiz 7: Sampling Distributions

Which of the following is another name for assignable cause variation?

The spread of the distribution of sample means is considerably less than the spread of the sampled population.

If random samples of size n = 36 are drawn from a non-normal population with finite mean = 75 and standard deviation = 15, then the sampling distribution of the sample mean is approximately normally distributed with mean = 75 and standard deviation = 2.5.

For a control chart, where are the lower and upper control limits usually set?

According to the Central Limit Theorem, for large samples, the standard error of the sample mean is the population standard deviation divided by the square root of the sample size.

If the standard error of the sampling distribution of the sample proportion is 0.02049 for samples of size 500, then what may we conclude about the population proportion?

What are some of the most common sources of random variation?

As a general rule, the normal distribution provides a good approximation to the sampling distribution of the sample proportion, , only if which of the following conditions holds?

Given a population proportion of p = 0.8 and a sample size of n = 100, what is the standard deviation of the sampling distribution of the sample proportion, ?