Question

A random sample of size \(n=40\) is selected from a population with mean \(\mu=100\) and standard deviation \(\sigma=20 .\) a. What will be the approximate shape of the sampling distribution of \(\bar{x} ?\) b. What will be the mean and standard deviation of the sampling distribution of \(\bar{x} ?\)

Short answer

Answer: The approximate shape of the sampling distribution of the sample mean is normal, the mean is 100, and the standard deviation is approximately 3.16.

Step-by-step solution

Identifying given data, and using the Central Limit Theorem to determine the approximate shape of the sampling distribution of 𝑥̄

The given data is: - Population mean (μ) = 100 - Population standard deviation (σ) = 20 - Sample size (n) = 40 According to the Central Limit Theorem, if the sample size (n) is large enough (which is typically n ≥ 30), the sampling distribution of the sample mean (𝑥̄) will be approximately normally distributed. Since n = 40, which is greater than 30, the approximate shape of the sampling distribution of 𝑥̄ will be approximately normally distributed. Answer (a): The approximate shape of the sampling distribution of 𝑥̄ is normal.

Finding the mean of the sampling distribution of 𝑥̄

To find the mean of the sampling distribution of 𝑥̄, we can use the formula: Mean of sampling distribution of 𝑥̄ = μ Given that μ = 100, we can find the mean of the sampling distribution of 𝑥̄ as: Mean of sampling distribution of 𝑥̄ = 100 Answer (b): The mean of the sampling distribution of 𝑥̄ is 100.

Finding the standard deviation of the sampling distribution of 𝑥̄

To find the standard deviation of the sampling distribution of 𝑥̄, we can use the formula: Standard deviation of the sampling distribution of 𝑥̄ = σ / √n Given that σ = 20 and n = 40, we can find the standard deviation of the sampling distribution of 𝑥̄ as: Standard deviation of the sampling distribution of 𝑥̄ = 20 / √40 Standard deviation of the sampling distribution of 𝑥̄ = 20 / 6.32 (approximately) Standard deviation of the sampling distribution of 𝑥̄ = 3.16 (approximately) Answer (b): The standard deviation of the sampling distribution of 𝑥̄ is approximately 3.16.

Key concepts

Sampling Distribution

In statistics, a sampling distribution is the probability distribution of a given statistic based on a random sample. It’s a fundamental concept because it underlies the rationale for inferential statistics, which helps us make predictions or inferential judgments about a population based on a sample.

The term "sampling distribution" is most often used in reference to the distribution of the sample mean. When you take a random sample from a population, the sample mean will vary from sample to sample. The sampling distribution is essentially a collection of these sample means.

One important note about the sampling distribution of the sample mean is that its shape becomes approximately normal as the sample size increases, due to the Central Limit Theorem. This assumes that your sample size is large enough, typically n ≥ 30, and that you're sampling from a population with a finite standard deviation.

Sample Mean

The sample mean is represented by the symbol \( \bar{x} \). It is the average of an individual sample and serves as an estimate of the population mean. Calculating the sample mean is straightforward: add up all the values in the sample and divide by the number of observations.

The sample mean plays a critical role in sampling distributions. As per the Central Limit Theorem, when you take the mean of all samples, the distribution of those sample means will approximate a normal distribution, especially as the sample size grows larger.

Notably, the mean of the sampling distribution of \( \bar{x} \) is equal to the population mean \( \mu \). This relationship provides insight into why we expect a sample mean, when taken from a large random sample, to be a good estimate of the population mean.

Standard Deviation of the Sampling Distribution

The standard deviation of the sampling distribution is often called the standard error and is a measure of how much the sample mean \( \bar{x} \) of different samples would differ from the actual population mean \( \mu \).

To compute this, we utilize the following formula: \[\text{Standard Deviation of the Sampling Distribution} = \frac{\sigma}{\sqrt{n}}\]Where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

In our exercise, it's calculated as \( 20 / \sqrt{40} \), which approximately equals 3.16. The smaller this value, the closer your sample mean is likely to be to the actual population mean. Thus, increasing the sample size leads to a smaller standard deviation of the sampling distribution, which implies more reliable estimates.

Normal Distribution

The normal distribution, sometimes called the "bell curve," is a type of continuous probability distribution for a real-valued random variable. Understanding normal distribution is critical in the context of the sampling distribution.

It is characterized by its symmetric bell shape, showing that data near the mean are more frequent in occurrence than data far from the mean. The Central Limit Theorem tells us that with a large enough sample size, the sampling distribution of the sample mean will be normally distributed, even if the original population distribution is not.

• The mean, median, and mode of a perfectly normal distribution are all the same, and it is fully determined by its mean and standard deviation.
• In practical terms, this property ensures that the methods of statistical inference remain robust to various distributions of the data — as long as the sample size is sufficiently large.

By understanding these concepts, students can better appreciate how the Central Limit Theorem supports the use of normal distribution in inferential statistics.