When dealing with a population that has an established mean ( extit{μ}) of 80 and a standard deviation ( extit{σ}) of 7, we can apply concepts of inferential statistics to understand how a sample of 49 observations behaves.
First, it’s important to note that when we take a sample from a population, the sample mean ( extit{X̄}) will tend to approximate the population mean as the sample size increases. Given our sample size of 49, which is relatively large, we can expect the sample mean to be close to 80.
Next, we need to consider the variability of the sample mean. The standard deviation of the sample mean (also known as the standard error) can be calculated using the formula:
Standard Error (SE) = σ / √nWhere:
- σ = population standard deviation (7)
- n = sample size (49)
So, substituting our values:
SE = 7 / √49 = 7 / 7 = 1This tells us that the sample mean will have a standard error of 1. This means that if we were to take many samples of size 49 from this population, the means of those samples would typically vary by about 1 unit around the population mean of 80.
Furthermore, assuming the population distribution is normal (or the sample size is large enough for the Central Limit Theorem to apply), we can also construct a confidence interval for the sample mean. For a 95% confidence level, we can use the formula:
Confidence Interval = X̄ ± Z * SEIn this case, the Z-value for a 95% confidence interval is approximately 1.96:
CI = 80 ± 1.96 * 1Which calculates to:
CI = [78.04, 81.96]This interval means we can be 95% confident that the true population mean lies between 78.04 and 81.96. Overall, this analysis provides valuable insights into what we can expect from our sample of 49 observations taken from a population with a mean of 80 and a standard deviation of 7.