Final answer:
To find the sample mean when you know the z-score, use the formula: Sample Mean = (Population Mean) + (Z-score) × (Standard Error of the Mean), where the Standard Error of the Mean equals the population's standard deviation divided by the square root of the sample size.
Step-by-step explanation:
The formula required to find the value of a sample mean when you know the z-score is Option D which is: Sample Mean = (Population Mean) + (Z-score) × (Standard Error of the Mean). The standard error of the mean is calculated by dividing the standard deviation of the population by the square root of the sample size (n). When you have a z-score, you are essentially being told how many standard errors a value is from the population mean. Therefore, knowing the population mean (μ), z-score (z), and standard error of the mean, you can rearrange the z-score formula to solve for the sample mean (x-bar).
To calculate the sample mean with a z-score, you'd use a calculator or computer to verify the mean and standard deviation. If you're working with a standard normal distribution (z-distribution), keep in mind that the population mean is 0 and the standard deviation is 1. However, when working with samples or different populations, you need to use the specific mean and standard deviation of that dataset along with the z-score to find the sample mean.
Let's consider a straightforward example where the population mean is 70, standard deviation is 9, and z-score is 1.5. First, calculate the standard error using the formula σ√n, say for a sample size of 60. Next, plug in the values: Sample Mean = 70 + (1.5 × (9√60)). This would give us the specific value for the sample mean that is 1.5 standard errors above the population mean.