A \(95 \%\) confidence interval for a population mean was reported to be 152 to \(160 .\) If \(\sigma=15\) what sample size was used in this study?

The population mean, often represented by the Greek letter \( \mu \), is a measure that reflects the average of a particular characteristic of an entire group or population. Picture a large group of individuals or items, and the population mean is the average value of a specific trait they all share, such as height, weight, or test score. When it's not feasible to measure every single entity in the population, a sample mean (\(\bar{X}\)) can be used as an estimated value. This is calculated by adding together all of the measured values from a sample (a smaller, selected group from the population) and dividing by the number of observations in the sample. In our exercise, the sample mean was used as a foundational figure in determining the confidence interval for the population mean.

Determining the correct sample size is crucial for any study, as it impacts the accuracy and reliability of the results. It's a balance between practical constraints, like cost and time, and the desired precision of the estimates. The larger the sample size, the closer we get to the actual population mean, but increasing sample size may involve higher costs and more time consuming data collection. The central question of sample size determination is figuring out how large a sample should be to accurately estimate the population parameters with a given level of confidence. It involves complex calculations that take into account the variability of the data (\( \sigma \), the standard deviation), the desired level of precision (margin of error), and the desired confidence level (which affects the Z-score).

Our exercise illustrates how we can work backwards: using the confidence interval and standard deviation to deduce the sample size employed in the study.

A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. In the context of confidence intervals, Z-scores are used to determine how far the margin of error extends from the mean. These scores come from the standard normal distribution. For a 95% confidence interval, the Z-score is typically 1.96, which signifies that the confidence interval extends 1.96 standard deviations on either side of the mean. The exercise used this Z-score to help calculate the sample size.

The standard normal distribution, also known as the Z-distribution, is a special case of the normal distribution that is centered at zero and has a standard deviation of one. It is used in various statistical analyses because of its property where areas under the curve correspond to percentiles. This means that we can calculate the probability of a random variable falling within a certain range. The Z-score is essentially the number of standard deviations away from the mean a data point is, and it corresponds to a certain area under the standard normal distribution curve. This concept is integral when calculating confidence intervals, as it allows us to specify the range within which we can expect the true population parameter to lie with a certain level of confidence.

The margin of error represents the extent of the range around the sample mean within which we can be relatively certain the true population mean lies. It's part of creating a confidence interval, which is an estimated range of values that is likely to include the population parameter. The margin of error is affected by the sample size and the variability in the data, as well as the confidence level one wishes to have in the interval estimate. In simpler terms, it tells us how 'wrong' our estimate could be. In the exercise, we calculated the margin of error using the given confidence interval and used it alongside the known standard deviation and Z-score to determine the required sample size.