7.9
Estimating Population Mean with Known Standard Deviation
To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an…
Consider an example wherein a truck container is to be redesigned to accommodate longer oakwood logs.
The container is designed based on obsolete measurements, so engineers require a new mean length of the logs.
As getting measurements of all the oakwood trees or logs in the world is impossible, samples can be drawn from the available stock.
This is the sample mean, which is the best point estimate of the population mean when its standard deviation is small.
However, the confidence interval can provide a more reliable estimate of the population mean, which requires calculating the margin of error using the following equation.
If the population and samples both assume the normal distribution, and the sample size is more than 30, a critical value can be obtained using the z distribution.
However, determining the population mean with these assumptions requires prior knowledge of population standard deviation, which is an unrealistic situation.
In the example of oakwood logs, previous forestry studies may provide this standard deviation to calculate the margin of error.
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Q1: Why is the sample mean considered the best point estimate of population mean?
The sample mean is the best point estimate of the unknown population mean when the population standard deviation is small and the sample is randomly drawn. It provides a single value estimate based on available data, though it becomes more reliable when combined with a confidence interval for a more complete picture of the population parameter.
Q2: What is the margin of error and how does it relate to confidence intervals?
The margin of error, also called the error bound for a population mean (EBM), quantifies the range of uncertainty around a point estimate. It depends on the confidence level and is used to construct the confidence interval by creating upper and lower bounds: point estimate minus error bound and point estimate plus error bound.
Q3: When can you use the z distribution to find critical values for population mean estimation?
You can use the z distribution to obtain critical values when both the population and sample follow a normal distribution and the sample size exceeds 30. This approach requires prior knowledge of the population standard deviation, which may come from previous studies or historical data.
Q4: What is the difference between confidence level and alpha in interval estimation?
The confidence level (CL) represents the percent of confidence intervals that contain the true population parameter when repeated samples are taken. Alpha (α) is the probability that the interval does not contain the population parameter. Mathematically, α and CL are complementary: α + CL = 1.
Q5: What are the steps to construct a confidence interval for a known population standard deviation?
First, calculate the sample mean from random sample data. Second, find the z-score corresponding to your chosen confidence level. Third, calculate the error bound using the margin of error formula. Finally, construct the confidence interval and write an interpretation in context of the problem.
Q6: How does sample size affect the reliability of population mean estimates?
Larger sample sizes improve the reliability of estimates because sample means follow an approximately normal distribution. When sample size exceeds 30, the z distribution can be reliably used for critical value determination, making the confidence interval more precise and trustworthy for estimating the true population mean.
Q7: Why is knowing the population standard deviation important for this estimation method?
The population standard deviation is essential because it directly affects the margin of error calculation, which determines the width of the confidence interval. Without this value, you cannot accurately quantify uncertainty around your estimate. When unknown, alternative methods like estimating population mean with unknown standard deviation must be used instead.