Question 1

Suppose ? = 50 and$\sigma$ = 10 for a population.In a sample where n = 100 is randomly taken,90% of all possible sample means will fall between 49 and 51.

Accepted Answer

The standard error of the mean is $\sigma/\sqrt{n} = 10/\sqrt{100} = 1$. For a 90% confidence interval, the z-score is approximately 1.645. The interval is $\mu \pm 1.645 \times 1$, which is 48.355 to 51.645, not 49 to 51.

Question 2

The Central Limit Theorem ensures that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, as long as the sample is random and the sample size is not too small.

Question 3

As the sample size increases,the effect of an extreme value on the sample mean becomes smaller.

Accepted Answer

As the sample size increases, the influence of extreme values gets diluted by the contribution of other values in the sample leading to a smaller effect on the sample mean. This is due to the fact that the sample mean is computed as the sum of all values in the sample divided by the sample size, so the larger the sample size, the smaller the impact of any individual value on the mean.

Question 4

If the amount of gasoline purchased per car at a large service station has a population mean of 15 gallons and a population standard deviation of 4 gallons and it is assumed that the amount of gasoline purchased per car is symmetric,there is approximately a 68.26% chance that a random sample of 16 cars will have a sample mean between 14 and 16 gallons.

Accepted Answer

The given statement is related to the central limit theorem which states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population distribution, as long as the sample size is sufficiently large. The sample mean has a standard error of (standard deviation of population) / sqrt(sample size) = 4 / sqrt(16) = 1. Therefore, the interval of (14, 16) is 1 standard error away from the population mean of 15. By the empirical rule, there is approximately a 68.26% chance that the sample mean falls within 1 standard error of the population mean. Therefore, the given statement is true.

Question 5

The fact that the sample means are less variable than the population data can be observed from the standard error of the mean.

Accepted Answer

This is true, as the standard error of the mean (SEM) measures the variability of the sample means from multiple samples taken from the same population. Generally, the SEM is smaller than the standard deviation of the population, indicating that the sample means are less variable than the population data.

Question 6

Suppose ? = 50 and $\sigma$ = 10 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall between 48.04 and 51.96.

Accepted Answer

This statement is true based on the given information and the properties of the normal distribution. The sample mean will have a normal distribution with mean μ and standard deviation σ/√n. Using the given values, we can find that this standard deviation is 10/√100 = 1. Therefore, the interval from 48.04 to 51.96 is a 95% confidence interval for the true population mean, meaning that we can be 95% confident that the true population mean falls in this range.

Question 7

If the population distribution is skewed,in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Accepted Answer

This is known as the Central Limit Theorem, which states that as sample size increases, the sampling distribution of the mean approaches a normal distribution regardless of the shape of the population distribution. However, this approximation becomes more accurate as the sample size increases beyond 30.

Question 8

Suppose ? = 80 and $\sigma$= 20 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall above 76.08.

Accepted Answer

This is true because the sample mean is an unbiased estimator of the population mean and the standard error of the mean (SE) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). Using the formula for the SE, SE = σ/√n = 20/√100 = 2. Therefore, a 95% confidence interval for the population mean would be 80 ± 1.96(2) or (75.16, 84.84). Since 76.08 > 75.16, it is true that 95% of all possible sample means will fall above 76.08.

Question 9

The amount of time required for an oil and filter change on an automobile is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes.A random sample of 16 cars is selected.What is the standard error of the mean?

Accepted Answer

The answer of The amount of time required for an...

Question 10

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.So,95% of all sample means will be greater than how many grams?

Accepted Answer

The answer of The amount of tea leaves in a...

Question 11

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be less than 100 grams?

Accepted Answer

The answer of The amount of tea leaves in a...

Question 12

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.So,the middle 70% of all sample means will fall between what two values?

Accepted Answer

The answer of The amount of tea leaves in a...

Question 13

For distributions such as the normal distribution,the arithmetic mean is considered more stable from sample to sample than other measures of central tendency.

Accepted Answer

This is true because the normal distribution is symmetric, and the arithmetic mean is calculated based on the values in the distribution, thereby providing a stable measure of central tendency. Other measures, such as the median and mode, may change significantly from sample to sample depending on the distribution of the data.

Question 14

A sampling distribution is defined as the probability distribution of possible sample sizes that can be observed from a given population.

Accepted Answer

A sampling distribution is actually the probability distribution of a sample statistic, such as the mean or standard deviation, that can be observed from multiple samples drawn from a given population.

Question 15

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be between 100 and 120 grams?

Accepted Answer

The answer of The amount of tea leaves in a...

Question 16

A sampling distribution is a distribution for a statistic.

Accepted Answer

A sampling distribution is a distribution of a statistic, such as the mean or standard deviation, that is calculated from samples taken from a population.

Question 17

If the amount of gasoline purchased per car at a large service station has a population mean of 15 gallons and a population standard deviation of 4 gallons and a random sample of 64 cars is selected,there is approximately a 95.44% chance that the sample mean will be between 14 and 16 gallons.

Accepted Answer

This is a statement about a confidence interval for the sample mean. Given a sample size of 64, the standard error of the mean is:

SE = σ/√n = 4/√64 = 0.5

To calculate the 95% confidence interval for the sample mean, we can use the formula:

CI = x̄ ± z*SE

where x̄ is the sample mean, z* is the z-score corresponding to the desired level of confidence (in this case, 1.96 for 95%), and SE is the standard error of the mean.

Substituting in the values given in the problem, we get:

CI = 15 ± 1.96*0.5
CI = (14, 16)

This means that we are 95% confident that the true population mean lies between 14 and 16 gallons. Therefore, the statement in the question is true.

Question 18

The standard error of the mean is also known as the standard deviation of the sampling distribution of the sample mean.

Accepted Answer

The standard error of the mean is indeed the standard deviation of the sampling distribution of the sample mean.

Question 19

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be greater than 100 grams?

Accepted Answer

The answer of The amount of tea leaves in a...

Question 20

As the size of the sample is increased,the standard deviation of the sampling distribution of the sample mean for a normally distributed population will stay the same.

Accepted Answer

As the size of the sample is increased, the standard deviation of the sampling distribution of the sample mean, also known as the standard error, decreases. This is because the standard error is inversely related to the square root of the sample size.

Quiz 7: Sampling Distributions

Suppose ? = 50 and $\sigma$ = 10 for a population.In a sample where n = 100 is randomly taken,90% of all possible sample means will fall between 49 and 51.

The Central Limit Theorem ensures that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

As the sample size increases,the effect of an extreme value on the sample mean becomes smaller.

The fact that the sample means are less variable than the population data can be observed from the standard error of the mean.

Suppose ? = 50 and $\sigma$ = 10 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall between 48.04 and 51.96.

If the population distribution is skewed,in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Suppose ? = 80 and $\sigma$ = 20 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall above 76.08.

The amount of time required for an oil and filter change on an automobile is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes.A random sample of 16 cars is selected.What is the standard error of the mean?

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.So,95% of all sample means will be greater than how many grams?

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be less than 100 grams?

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.So,the middle 70% of all sample means will fall between what two values?

For distributions such as the normal distribution,the arithmetic mean is considered more stable from sample to sample than other measures of central tendency.

A sampling distribution is defined as the probability distribution of possible sample sizes that can be observed from a given population.

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be between 100 and 120 grams?

A sampling distribution is a distribution for a statistic.

If the amount of gasoline purchased per car at a large service station has a population mean of 15 gallons and a population standard deviation of 4 gallons and a random sample of 64 cars is selected,there is approximately a 95.44% chance that the sample mean will be between 14 and 16 gallons.

The standard error of the mean is also known as the standard deviation of the sampling distribution of the sample mean.

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be greater than 100 grams?

As the size of the sample is increased,the standard deviation of the sampling distribution of the sample mean for a normally distributed population will stay the same.

Quiz 7: Sampling Distributions

Suppose ? = 50 and σ\sigmaσ = 10 for a population.In a sample where n = 100 is randomly taken,90% of all possible sample means will fall between 49 and 51.

The Central Limit Theorem ensures that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

As the sample size increases,the effect of an extreme value on the sample mean becomes smaller.

The fact that the sample means are less variable than the population data can be observed from the standard error of the mean.

Suppose ? = 50 and σ\sigmaσ = 10 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall between 48.04 and 51.96.

If the population distribution is skewed,in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Suppose ? = 80 and σ\sigmaσ = 20 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall above 76.08.

The amount of time required for an oil and filter change on an automobile is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes.A random sample of 16 cars is selected.What is the standard error of the mean?

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.So,95% of all sample means will be greater than how many grams?

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be less than 100 grams?

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.So,the middle 70% of all sample means will fall between what two values?

For distributions such as the normal distribution,the arithmetic mean is considered more stable from sample to sample than other measures of central tendency.

A sampling distribution is defined as the probability distribution of possible sample sizes that can be observed from a given population.

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be between 100 and 120 grams?

A sampling distribution is a distribution for a statistic.

If the amount of gasoline purchased per car at a large service station has a population mean of 15 gallons and a population standard deviation of 4 gallons and a random sample of 64 cars is selected,there is approximately a 95.44% chance that the sample mean will be between 14 and 16 gallons.

The standard error of the mean is also known as the standard deviation of the sampling distribution of the sample mean.

The amount of tea leaves in a can from a production line is normally distributed with μ = 110 grams and σ = 25 grams.A sample of 25 cans is to be selected.What is the probability that the sample mean will be greater than 100 grams?

As the size of the sample is increased,the standard deviation of the sampling distribution of the sample mean for a normally distributed population will stay the same.

Suppose ? = 50 and $\sigma$ = 10 for a population.In a sample where n = 100 is randomly taken,90% of all possible sample means will fall between 49 and 51.

Suppose ? = 50 and $\sigma$ = 10 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall between 48.04 and 51.96.

Suppose ? = 80 and $\sigma$ = 20 for a population.In a sample where n = 100 is randomly taken,95% of all possible sample means will fall above 76.08.