Sampling Distribution of the Sample Mean and Central Limit Theorem: Videos & Practice Problems

To estimate a population mean, we often rely on the sample mean from a random sample, but this can lead to sampling error. The sampling distribution of the sample mean, which becomes approximately normal as sample size increases (Central Limit Theorem), provides a more reliable estimate. For sample sizes of 30 or more, we can calculate probabilities using z-scores. The standard deviation of the sampling distribution is smaller than the population's, indicating less variability among sample means, enhancing prediction accuracy for the population mean.

Sampling Distribution of Sample Mean

To estimate the mean of a population, collecting data from every member is often impractical. Instead, we typically gather data from a random sample and use the sample mean as an approximation of the population mean. However, this method carries the risk of obtaining a non-representative sample, which can lead to inaccurate estimates. To mitigate this risk, we can collect multiple samples of the same size and analyze the distribution of their means, known as the sampling distribution of the sample mean (denoted as \( \bar{x} \)).

The sampling distribution provides a frequency distribution of sample means, allowing us to observe which means are more common. For instance, if a pet store wants to estimate the average number of pets owned by American households, they might first take a single sample of 30 individuals, yielding a sample mean. However, to improve accuracy, they could take multiple samples (e.g., 10 samples of 30) and calculate the average of these sample means. This average is likely to be a better predictor of the population mean.

In practice, if one sample yields a mean of 4 pets, it may not accurately reflect the population mean, which could be closer to 2 or 3 pets. Variability in sample means can occur due to the randomness of sampling. By analyzing a sampling distribution, we can find that the average of the sample means might be around 2.55, aligning more closely with the expected population mean. This demonstrates that while a single sample can be misleading, averaging multiple samples tends to yield a more reliable estimate.

Moreover, the shape of the sampling distribution often approaches a normal distribution, even if the underlying population distribution is skewed. This phenomenon is explained by the Central Limit Theorem, which states that as the sample size increases, the distribution of the sample means will tend to become normal, regardless of the original distribution of the population. For example, with a sample size of 30, the distribution of sample means starts to resemble a bell curve, with most means clustering around the population mean and fewer extreme values.

This property of sampling distributions is significant because it allows statisticians to apply the principles of normal distribution to make inferences about the population mean. Understanding the behavior of sampling distributions and the Central Limit Theorem equips us with powerful tools for statistical analysis, enabling more accurate predictions and conclusions based on sample data.

For a new advertising campaign, a video game retailer is interested in including information on the average play time of their most popular game. They get 100 random samples of 40 players and obtain their play time to get a sampling distribution. The mean of the sampling distribution is 26.7 hours. In this example, what is the value of $n$?

• A. 100
• B. 40
• C. 27
• D. Unknown

Answer: (B)

A researcher takes 10 samples of 20 students each to get a sampling distribution of the average number of siblings students at a university have. According to the Central Limit Theorem, what can the researcher do make their sampling distribution get closer to normal?

• A. Increase the number of samples
• B. Increase the sample size
• C. Decrease sample size
• D. Decrease number of samples

Answer: (B)

Central Limit Theorem

The central limit theorem (CLT) is a fundamental concept in statistics that states that for any random variable, regardless of its distribution, as the sample size (n) increases, the sampling distribution of the sample mean (\( \bar{x} \)) approaches a normal distribution. This is significant because it allows us to apply the properties of the normal distribution to make inferences about the sample mean, even when the original data does not follow a normal distribution.

According to the CLT, when the sample size \( n \) is 30 or greater, we can assume that the sampling distribution of the sample mean is approximately normal. This enables us to calculate z-scores and probabilities using statistical tools or z-tables. For instance, if we roll a die 30 times and repeat this experiment to obtain 50 samples, we can analyze the resulting sampling distribution of the sample mean.

In this scenario, the expected population mean (\( \mu_x \)) for a die roll is 3.5, and the population standard deviation (\( \sigma_x \)) is approximately 1.71. Since our sample size \( n \) is 30, we can confidently apply the CLT. The mean of the sampling distribution (\( \mu_{\bar{x}} \)) will equal the population mean, while the standard deviation of the sampling distribution (\( \sigma_{\bar{x}} \)) is calculated using the formula:

\( \sigma_{\bar{x}} = \frac{\sigma_x}{\sqrt{n}} \)

This formula indicates that the standard deviation of the sampling distribution will be smaller than the population standard deviation, reflecting less variation between sample means as the sample size increases.

To find the probability of obtaining a sample mean less than 2.5, we first calculate the z-score using the formula:

\( z = \frac{\bar{x} - \mu_x}{\sigma_{\bar{x}}} \)

Substituting the values, we have:

\( z = \frac{2.5 - 3.5}{\frac{1.71}{\sqrt{30}}} \)

Calculating this gives a z-score of approximately -3.2. We can then use this z-score to find the corresponding probability. The probability that \( \bar{x} \) is less than 2.5 is equivalent to finding the probability of a z-score less than -3.2, which yields a very small probability of approximately 0.0007. This low probability indicates that obtaining a sample mean of 2.5 from 30 rolls is quite unlikely, given that the population mean is 3.5.

Understanding the central limit theorem and its application to sampling distributions is crucial for making informed statistical inferences. For further practice with the CLT and sampling distributions, exploring additional examples can enhance comprehension and application of these concepts.

A company’s marketing team takes 50 samples of 10 recent clients to create a sampling distribution of sample means for the average amount spent per month on company products. Can the Central Limit Theorem be used to determine that the sampling distribution is normal?

• A. No
• B. Yes
• C. More information is required

Answer: (A)

If ${\mu }_X=3.2$ and ${\sigma }_X=0.98$, find the probability of getting a sample mean above 3.5 in a sample of 60 people.

• A. 0.009
• B. 0.018
• C. 0.003
• D. 0.006

Answer: (A)

Central Limit Theorem Example 1

In this scenario, we are tasked with determining the probability that a random sample of 36 moviegoers will give an upcoming film an average rating of 4.3 stars or higher. The average score from previous test audiences is 4.2 stars, with a standard deviation of 0.25 stars. To solve this, we will utilize the Central Limit Theorem, which states that for a sufficiently large sample size (n ≥ 30), the sampling distribution of the sample mean will be approximately normally distributed.

Given that our sample size, n, is 36, we can proceed with the calculations. The population mean (μ) is 4.2 stars, and the population standard deviation (σ) is 0.25 stars. We are interested in finding the probability that the sample mean (x̄) is greater than or equal to 4.3 stars. This can be expressed as:

\[ P(x̄ \geq 4.3) \]

To find this probability, we first calculate the z-score using the formula:

\[ z = \frac{x̄ - μ}{\sigma / \sqrt{n}} \]

Substituting the known values:

\[ z = \frac{4.3 - 4.2}{0.25 / \sqrt{36}} \]

Calculating the denominator:

\[ \sigma / \sqrt{n} = 0.25 / 6 = 0.04167 \]

Now substituting back into the z-score formula gives:

\[ z = \frac{0.1}{0.04167} \approx 2.4 \]

Next, we need to find the probability associated with this z-score. Since we are looking for the probability that the z-score is greater than or equal to 2.4, we will refer to the standard normal distribution table or use technology to find this area. The area to the right of z = 2.4 corresponds to a probability of approximately 0.008.

Thus, the probability that a random sample of 36 moviegoers will give the film an average rating of 4.3 stars or greater is approximately 0.008, indicating that it is quite unlikely for this to occur.

Here’s what students ask on this topic:

What is the Central Limit Theorem and why is it important in statistics?

The Central Limit Theorem (CLT) states that for any random variable, regardless of its original distribution, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases. This is significant because it allows statisticians to make inferences about population parameters using the properties of the normal distribution. For sample sizes of 30 or more, the CLT ensures that the sampling distribution is approximately normal, enabling the use of z-scores and probability calculations. This theorem is foundational in statistics as it justifies the use of normal distribution techniques in a wide range of applications, even when the population distribution is not normal.

How does the sample size affect the sampling distribution of the sample mean?

The sample size has a significant impact on the sampling distribution of the sample mean. As the sample size increases, the sampling distribution becomes narrower, meaning the standard deviation of the sampling distribution (also called the standard error) decreases. This is because the standard error is calculated as the population standard deviation divided by the square root of the sample size:

$\sigma$_x=σn

Additionally, as the sample size increases, the sampling distribution becomes more closely approximated by a normal distribution, regardless of the population's original distribution, due to the Central Limit Theorem.

What is the difference between the population mean and the sample mean?

The population mean, denoted as $\mu$, is the average of all data points in an entire population. It is a fixed value that represents the true central tendency of the population. The sample mean, denoted as $\overline{x}$, is the average of data points in a random sample taken from the population. It is used as an estimate of the population mean. However, the sample mean may vary from sample to sample due to sampling variability. The sampling distribution of the sample mean helps us understand how close the sample mean is likely to be to the population mean.

How do you calculate the standard deviation of the sampling distribution of the sample mean?

The standard deviation of the sampling distribution of the sample mean, also known as the standard error, is calculated using the formula:

$\sigma$_x=σn

Here, $\sigma$ is the population standard deviation, and $n$ is the sample size. This formula shows that as the sample size increases, the standard error decreases, meaning the sample means are more tightly clustered around the population mean.

When can the Central Limit Theorem be applied to a sampling distribution?

The Central Limit Theorem can be applied to a sampling distribution when the sample size is sufficiently large, typically $n\ge 30$. For smaller sample sizes, the theorem may still apply if the population distribution is already normal. However, for non-normal population distributions, a larger sample size is required to ensure the sampling distribution of the sample mean approximates a normal distribution. The CLT is crucial because it allows the use of normal distribution techniques to analyze sample means, even when the population distribution is unknown or non-normal.

Why is the sampling distribution of the sample mean narrower than the population distribution?

The sampling distribution of the sample mean is narrower than the population distribution because the variability between sample means is less than the variability between individual data points in the population. The standard deviation of the sampling distribution, called the standard error, is calculated as:

$\sigma$_x=σn

Here, dividing the population standard deviation by the square root of the sample size reduces the variability. As the sample size increases, the standard error decreases, making the sampling distribution narrower and the sample mean a more precise estimate of the population mean.