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 be unreliable. The sampling distribution of the sample mean, $\left(x\right)$, provides a more accurate approximation by averaging multiple sample means. The Central Limit Theorem states that as sample size increases (n ≥ 30), the sampling distribution approaches a normal distribution, allowing for the use of z-scores and probabilities to make inferences about 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 \( \bar{x} \)). This distribution provides a clearer picture of how sample means vary and helps us predict the population mean more reliably.

For example, consider a pet store aiming to determine the average number of pets owned by American households. They might first take a single sample of 30 individuals, yielding a sample mean of 4. However, this mean could be misleading if the sample is not representative of the broader population, which might actually have a mean closer to 2 or 3 pets. By taking multiple samples, say 10 samples of 30 individuals each, the pet store can create a sampling distribution and calculate the average of these sample means. This average, in this case, might be 2.55, which aligns more closely with the expected population mean.

The reliability of the sampling distribution stems from the principle that while individual sample means can vary significantly, averaging multiple sample means tends to yield a result that is closer to the true population mean. This is particularly true when the sample size is sufficiently large, as larger samples are more likely to represent the population accurately. The central limit theorem supports this concept, stating that regardless of the original distribution of the population, the distribution of the sample means will approach a normal distribution as the sample size increases. This means that with larger samples, the sampling distribution will exhibit a bell curve shape, with most sample means clustering around the population mean.

In summary, using a sampling distribution to estimate the population mean is a more reliable method than relying on a single sample. The central limit theorem assures us that as we increase our sample sizes, the distribution of sample means will become more normal, allowing us to apply statistical methods that leverage the properties of the normal distribution. This understanding is crucial for making informed predictions about population parameters 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 as the sample size (n) increases, the sampling distribution of the sample mean (\( \bar{x} \)) approaches a normal distribution, regardless of the original distribution of the data. This is particularly significant because it allows us to apply the properties of the normal distribution to make inferences about the sample mean, even when the underlying data is not normally distributed.

According to the CLT, when the sample size 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 standard statistical methods. 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 compared to individual data points. As the sample size increases, this standard deviation decreases, leading to more precise estimates of the population mean.

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 yields a z-score of approximately -3.2. We can then use a z-table or statistical software to find the probability associated with this z-score. The resulting probability of obtaining a sample mean less than 2.5 is approximately 0.0007, indicating that such an outcome is quite unlikely given the population mean of 3.5.

This example illustrates the power of the central limit theorem in statistical analysis, allowing us to make informed predictions about sample means based on the properties of the normal distribution.

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:

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

Substituting the known values:

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

Calculating the denominator:

\[ \frac{0.25}{\sqrt{36}} = \frac{0.25}{6} = 0.04167 \]

Now substituting back into the z-score formula:

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

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 perform a right-tailed test. Using a z-table or statistical software, we find that:

\[ P(z \geq 2.4) \approx 0.008 \]

This result indicates that there is approximately a 0.8% chance that a random sample of 36 moviegoers will give the film an average rating of 4.3 stars or greater. This low probability suggests that such a high average rating is unlikely based on the current audience ratings.

Here’s what students ask on this topic:

What is the sampling distribution of the sample mean, and why is it important?

The sampling distribution of the sample mean is the probability distribution of sample means obtained from all possible random samples of a specific size (n) from a population. It is important because it allows us to make inferences about the population mean without needing to survey the entire population. By analyzing the sampling distribution, we can estimate the population mean more reliably, especially when the sample size is large. Additionally, the Central Limit Theorem ensures that as the sample size increases (n ≥ 30), the sampling distribution approaches a normal distribution, even if the population distribution is not normal. This property enables us to use z-scores and probabilities to calculate confidence intervals and test hypotheses about the population mean.

What does the Central Limit Theorem state, and how is it applied 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 (n) increases, provided n ≥ 30. This theorem is crucial in statistics because it allows us to use the properties of the normal distribution to make inferences about population parameters. For example, we can calculate z-scores using the formula:

$\frac{x¯-\mu }{\frac{\sigma }{\sqrt{n}}}$

where $x¯$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and n is the sample size. This enables us to calculate probabilities and confidence intervals for the sample mean, even when the population distribution is unknown.

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

Sample size has a significant impact on the shape of the sampling distribution. According to the Central Limit Theorem, as the sample size (n) increases, the sampling distribution of the sample mean becomes more normal in shape, regardless of the population's original distribution. For small sample sizes, the sampling distribution may retain the skewness or irregularities of the population distribution. However, when n ≥ 30, the sampling distribution typically resembles a bell-shaped curve (normal distribution). Additionally, larger sample sizes reduce the standard deviation of the sampling distribution, calculated as:

$\frac{\sigma }{\sqrt{n}}$

This means that larger samples lead to less variability between sample means, making the sampling distribution narrower and more concentrated around the population mean.

Why is the sample mean a better predictor of the population mean when using a sampling distribution?

The sample mean becomes a better predictor of the population mean when using a sampling distribution because it accounts for variability across multiple samples. By averaging the means of many random samples, the effects of outliers or unrepresentative samples are minimized. This process reduces the likelihood of extreme deviations from the true population mean. Additionally, the Central Limit Theorem ensures that the sampling distribution of the sample mean is centered around the population mean, and its standard deviation decreases as the sample size increases. This makes the average of sample means a more reliable and accurate estimate of the population mean compared to using a single sample 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:

$\frac{\sigma }{\sqrt{n}}$

where $\sigma$ is the population standard deviation, and $n$ is the sample size. This formula shows that the standard error decreases as the sample size increases, meaning that larger samples result in less variability between sample means. This reduction in variability makes the sampling distribution narrower and more concentrated around the population mean, improving the reliability of the sample mean as an estimate of the population mean.