8.1: The Sampling Distribution of the sample mean

Question 1

What is the sampling distribution of the sample mean?

Answer 1

The sampling distribution of the sample mean is the probability distribution of all possible sample means that could be obtained from all possible samples of the same size from a population.

Question 2

What does the sample mean (x̄) signify in the context of sampling distribution?

Answer 2

The sample mean (x̄) is a point estimate of the population mean (μ) derived from a sample. It is used for making statistical inferences about the population mean.

Question 3

How is the mean of the sample mean distribution calculated from individual sample means?

Answer 3

To calculate the mean of the sample mean distribution, sum all the individual sample means and divide by the number of samples.

Question 4

What is the relationship between a point estimate and the true population mean?

Answer 4

A point estimate is a single-value estimate of a population parameter, such as the mean. The point estimate of the sample mean (x̄) can differ from the true population mean (μ), demonstrating the estimation error.

Question 5

How is the probability of a specific sample mean determined in a sampling distribution?

Answer 5

The probability of a specific sample mean is determined by its frequency among all possible samples. This is calculated by dividing the number of times the specific mean appears by the total number of different possible samples.

Question 6

How is the sample mean of two cars calculated in the car mileage example?

Answer 6

In the car mileage example, the sample mean is calculated by summing the mileages of the two cars and dividing by two. For example, if the mileages are 30 mpg and 32 mpg, the sample mean would be (30 + 32) / 2 = 31 mpg.

Question 7

How is the mean of the population of car mileages calculated in the given example?

Answer 7

The mean of the population of car mileages is computed by summing all the individual car mileages and dividing by the number of cars. With car mileages of 29, 30, 31, 32, 33, and 34 mpg, the population mean is (29 + 30 + 31 + 32 + 33 + 34) / 6 = 31.5 mpg.

Question 8

What is the main purpose of the sampling distribution of the sample mean?

Answer 8

The main purpose of the sampling distribution of the sample mean is to show how accurate the sample mean is likely to be as a point estimate of the population mean.

Question 9

What does it mean when a sample mean is called an “unbiased point estimate” of the population mean?

Answer 9

A sample mean is called an unbiased point estimate of the population mean if, on average, the sampling distribution of the sample mean is centered around the true population mean.

This means there is no systematic tendency for the sample mean to overestimate or underestimate the population mean.

Question 10

What is the relationship between the standard deviation of the population of all possible sample means (σx̄) and the standard deviation of the population (σ)?

Answer 10

The standard deviation of the population of all possible sample means (σx̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). This is represented as σx̄ = σ / √n.

Question 11

When does the distribution of the population of all possible sample means approximate a normal distribution?

Answer 11

The distribution of the population of all possible sample means approximates a normal distribution if the sampled population itself has a normal distribution, regardless of the sample size.

Question 12

How does the sample size affect the standard deviation of the sampling distribution (σx̄)?

Answer 12

As the sample size (n) increases, the standard deviation of the sampling distribution (σx̄) decreases. This is because larger samples tend to average out extreme values, making the distribution of sample means more closely clustered around the population mean.

Question 13

Why is the standard deviation of the sampling distribution (σx̄) important?

Answer 13

The standard deviation of the sampling distribution (σx̄) is important because it helps to understand how much the sample mean will vary from the population mean, which in turn aids in determining the appropriate sample size for a given level of precision.

Question 14

What is the significance of the formula σx̄ = σ / √n in the context of sampling distributions?

Answer 14

The formula σx̄ = σ / √n indicates that the variability of sample means decreases as the sample size increases, which shows that larger samples are more likely to yield a sample mean close to the population mean.

Question 15

How does increasing the sample size affect the standard deviation of the sampling distribution?

Answer 15

Increasing the sample size reduces the standard deviation of the sampling distribution.

This is because the standard deviation of the sample mean (σx̄) is equal to the population standard deviation (σ) divided by the square root of the sample size (n). So as n increases, σx̄ decreases.

Question 16

What does the Empirical Rule say about the distribution of sample means?

Answer 16

The Empirical Rule states that for a normal distribution, 95.44% of all possible sample means are within plus or minus two standard deviations (2σx̄) of the population mean (μ).

Question 17

If the sample size is n = 50 and the standard deviation of the population is 0.8, what is the standard deviation of the sampling distribution?

Answer 17

If n = 50 and σ = 0.8, then the standard deviation of the sampling distribution (σx̄) would be 0.8 / √50, which equals approximately 0.113.

Question 18

Using the Empirical Rule, what percentage of sample means would fall within plus or minus 0.226 mpg of the population mean when n = 50?

Answer 18

When n = 50, 95.44% of the sample means would fall within plus or minus two standard deviations of the population mean, which is plus or minus 0.226 mpg (2 x 0.113).

Question 19

Why is it significant that the larger sample of size n = 50 is more likely to give a sample mean closer to μ?

Answer 19

A larger sample size is significant because it yields a smaller standard deviation of the sampling distribution, which means the sample means are more closely clustered around the population mean.

This increases the likelihood that the sample mean will be a more accurate estimate of the population mean.

Question 20

What is the probability of observing a sample mean greater than or equal to 31.56 mpg if the population mean is exactly 31 mpg?

Answer 20

To find this probability, you would use the z-score formula. The probability P(x̄ ≥ 31.56) when μ = 31 and σx̄ = .113 is found by calculating P(z ≥ (31.56 - 31) / .113), which gives P(z ≥ 4.96). This probability is extremely low, less than 0.00003.

Question 21

What conclusion can be drawn if the probability of observing a sample mean as large as the one observed is very low?

Answer 21

If the probability of observing a sample mean as large as the observed one is very low, it implies strong evidence that the true population mean is different (likely larger) than the hypothesized value.

In statistical inference, this could lead to rejecting a null hypothesis or confirming the validity of an alternative hypothesis.

Question 22

What does the Central Limit Theorem (CLT) state about sampling distributions?

Answer 22

The Central Limit Theorem states that if the sample size n is sufficiently large, the sampling distribution of the sample mean x̄ will be approximately normally distributed, regardless of the shape of the population distribution.

This holds true even if the sampled population is not normally distributed.

Question 23

Does the Central Limit Theorem apply only when the population is normally distributed?

Answer 23

No, the Central Limit Theorem applies to any population distribution, whether it is normal or not. If the sample size is large enough, the sampling distribution of the sample mean will tend to be normal.

Question 24

How does the sample size affect the distribution of the sample mean according to the CLT?

Answer 24

According to the CLT, the larger the sample size n, the more closely the sampling distribution of the sample mean will resemble a normal distribution.

Question 25

What is the significance of the Central Limit Theorem in statistical analysis?

Answer 25

The Central Limit Theorem is significant because it allows statisticians to make inferences about population parameters using the sample mean, even when the population distribution is unknown or not normal.

Question 26

How large must the sample size be for the CLT to hold?

Answer 26

While the CLT begins to apply with smaller sample sizes, a common rule of thumb is that the sample size should be at least 30 for the CLT to hold well, especially for populations that are not significantly skewed.

Question 27

If the population standard deviation σ is known, what is the standard deviation of the sampling distribution of x̄ according to the CLT?

Answer 27

The standard deviation of the sampling distribution of x̄ (σx̄) is equal to σ divided by the square root of the sample size n (σx̄ = σ / √n).

Question 28

What can be inferred if a sample mean is significantly different from the hypothesized population mean?

Answer 28

If a sample mean is significantly different from the hypothesized population mean, it can provide strong evidence against the null hypothesis. In the context of the example provided, it can indicate that the new billing system has effectively reduced the mean bill payment time.

Question 29

What is a sample statistic?

Answer 29

A sample statistic is any descriptive measure of the sample measurements, such as the sample mean, sample median, sample variance, or sample standard deviation.

Question 30

What is meant by the sampling distribution of a sample statistic?

Answer 30

The sampling distribution of a sample statistic is the probability distribution that shows all possible values the sample statistic can take based on different possible samples from the population.

Question 31

What characterizes a sample statistic as an unbiased point estimate of a population parameter?

Answer 31

A sample statistic is an unbiased point estimate of a population parameter if the average of all possible values of the statistic equals the population parameter.

Question 32

Why is the sample mean considered an unbiased point estimate of the population mean?

Answer 32

The sample mean is considered an unbiased point estimate of the population mean because the average of all possible sample means is equal to the population mean.

Question 33

Is the sample variance s^2 an unbiased estimate of the population variance σ^2?

Answer 33

When the population is infinite, the sample variance s^2 is an unbiased estimate of the population variance σ^2. For finite populations, it is considered approximately unbiased.

Question 34

What is the finite population multiplier and when do you apply it?

Answer 34

The finite population multiplier is applied to the standard deviation of the sample mean when sampling from a finite population without replacement. It adjusts the standard deviation for the finite population size.

Question 35

How do you calculate the standard deviation of the sample mean for a finite population?

Answer 35

For a finite population, the standard deviation of the sample mean is calculated as σ / √n multiplied by the square root of (N - n) / (N - 1), where N is the population size and n is the sample size.