Enter the population mean, standard deviation, sample size, and target sample mean to calculate the probability of obtaining that sample mean in a normal distribution.
Probability of Obtaining the Sample Mean:
Understanding Sampling Distribution
In a normal distribution, the sampling distribution of the sample mean describes the probabilities of different possible values for the sample mean. The formula for calculating the probability of a given sample mean is:
$$ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} $$
where:
- \(\overline{X}\) = Target sample mean
- \(\mu\) = Population mean
- \(\sigma\) = Population standard deviation
- \(n\) = Sample size
Real-Life Example: Quality Control in Manufacturing
Suppose a manufacturer produces components with an average length of 100 mm and a standard deviation of 15 mm. If a random sample of 30 components is taken, what is the probability that the sample mean length will be 102 mm or greater?
Here’s how the calculation works:
- Step 1: Set the values: population mean (μ) = 100, population standard deviation (σ) = 15, sample size (n) = 30, and target sample mean (\(\overline{X}\)) = 102.
- Step 2: Calculate the standard error of the mean:
$$ \text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{30}} \approx 2.74 $$
- Step 3: Find the z-score:
$$ Z = \frac{102 - 100}{2.74} \approx 0.73 $$
- Step 4: Calculate the probability based on the z-score to find the likelihood of obtaining this sample mean or greater.