Distribution of Sample Mean - Excel: Videos & Practice Problems

Understanding sampling distributions is essential for calculating the probability of obtaining a sample mean above or below a specific value. When dealing with large sample sizes (typically n ≥ 30), the sampling distribution of the sample mean approximates a normal distribution. This allows us to use the normal distribution functions in Excel to find these probabilities efficiently.

To find the probability that a sample mean is less than a certain value (a left tail probability), the =NORM.DIST function in Excel is highly useful. This function requires four inputs: the value of interest (x̄, the sample mean), the mean of the sampling distribution (μ, the population mean), the standard deviation of the sampling distribution (σ/√n), and a logical value indicating whether to calculate the cumulative distribution (TRUE for cumulative probability).

For example, if a company produces soda bottles with a population mean volume of 16.75 fluid ounces and a population standard deviation of 0.43 fluid ounces, and a quality control officer samples 40 bottles, the sampling distribution’s standard deviation is calculated as:

After calculating the sample mean (x̄) from the data using Excel’s =AVERAGE() function, you can input these values into =NORM.DIST(x̄, \mu, \sigma_{\bar{x}}, TRUE) to find the probability that a second sample will have a mean less than x̄.

To find the probability that a sample mean is greater than a certain value (a right tail probability), you can use the complement rule. Since =NORM.DIST only calculates left tail probabilities directly, subtracting the left tail probability from 1 gives the right tail probability:

This approach allows you to determine the likelihood of obtaining a sample mean above a specified threshold.

By combining these Excel functions with the fundamental concepts of sampling distributions, you can efficiently analyze probabilities related to sample means. This method is particularly useful in quality control and other fields where understanding variability and probability in sample data is crucial.

When working with large datasets that lack detailed labels or cosmetic features, it is essential to maintain organization by clearly labeling data columns and calculations. This approach helps in tracking the analysis process and understanding the results when revisiting the data. In statistical analysis, when given multiple samples, such as 100 random samples each containing 50 clients, the mean of the sampling distribution can be used as an estimate for the true population mean. For example, if the mean job search duration across samples is calculated using the average function in Excel, this value serves as an estimate of the population mean for job search duration. Similarly, the mean salary across samples can be estimated in the same way.

To calculate probabilities related to sample means, the normal distribution function NORM.DIST in Excel is highly useful. When the population standard deviation (σ) is known, the standard deviation of the sampling distribution (also called the standard error) is calculated by dividing σ by the square root of the sample size (n), expressed as:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

For instance, if the population standard deviation for job search duration is 5 weeks and the sample size is 50, the standard error becomes \( \frac{5}{\sqrt{50}} \). Using this, the probability that a randomly selected sample has a mean below a certain value (e.g., 15 weeks) can be found by applying the cumulative distribution function:

\[ P(\bar{X} < x) = \text{NORM.DIST}(x, \mu, \sigma_{\bar{x}}, \text{TRUE}) \]

where \(x\) is the sample mean threshold, \(\mu\) is the mean of the sampling distribution, and \(\sigma_{\bar{x}}\) is the standard error. A very small probability indicates that such a sample mean is unlikely given the population parameters and sample size.

When calculating the probability that a sample mean exceeds a certain value, the complement rule is applied. This involves subtracting the cumulative probability up to that value from 1, as shown:

\[ P(\bar{X} > x) = 1 - P(\bar{X} \leq x) = 1 - \text{NORM.DIST}(x, \mu, \sigma_{\bar{x}}, \text{TRUE}) \]

For example, if the population standard deviation for salary is \$15,000 and the sample size is 50, the standard error is \( \frac{15,000}{\sqrt{50}} \). Using this, the probability that a sample mean salary exceeds \$64,500 can be calculated, often resulting in a high probability if the threshold is below the mean of the sampling distribution.

Overall, understanding how to estimate population means from sampling distributions and calculate probabilities using the normal distribution function is crucial for analyzing sample data effectively. These methods allow for informed conclusions about population parameters based on sample statistics, even when working with large and minimally labeled datasets.

An airport collected data from 80 samples, each consisting of 45 people, recording the mean distance flown in miles over the past year. These 80 sample means create a sampling distribution for the mean distance flown. To estimate the true population mean distance flown, the average of these sample means is calculated, resulting in an estimate of 1,775 miles. This approach leverages the principle that the mean of the sampling distribution serves as an unbiased estimator of the population mean.

Given prior information that the population standard deviation (σ) for distance flown is 451 miles, the next step involves calculating the probability that a randomly selected sample mean falls between 1,770 and 1,776 miles. Since the sampling distribution of the sample mean is approximately normal, the standard deviation of the sampling distribution, known as the standard error, is computed by dividing the population standard deviation by the square root of the sample size (n = 45):

\[ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{451}{\sqrt{45}} \]

To find the probability that the sample mean lies between two values, the cumulative distribution function (CDF) of the normal distribution is used. The function calculates the left-tail probabilities (probabilities that the sample mean is less than a given value) for both 1,776 and 1,770 miles. The probability of the sample mean falling between these two values is the difference between these left-tail probabilities:

\[ P(1770 < \bar{x} < 1776) = P(\bar{x} < 1776) - P(\bar{x} < 1770) \]

Using the normal distribution with mean 1,775 and the calculated standard error, the left-tail probabilities are approximately 0.54 for 1,776 miles and 0.309 for 1,770 miles. Subtracting these yields a probability of about 0.231, indicating a 23.1% chance that a randomly selected sample mean falls within this range.

This process highlights the application of the Central Limit Theorem, which ensures that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, allowing for probability calculations using the normal distribution. Understanding how to compute and interpret the standard error and use cumulative probabilities is essential for making inferences about population parameters based on sample data.