How large a sample should be selected to provide a \(95 \%\) confidence interval with a margin of error of \(10 ?\) Assume that the population standard deviation is 40

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. The range is calculated from a given set of sample data, primarily by using the sample mean and standard deviation.

In the context of the exercise, a 95% confidence interval means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect about 95 of those intervals to contain the population mean. It’s important to note that the confidence level doesn’t reflect the probability that a particular interval captures the mean, but rather the proportion of intervals that will capture the mean over a large number of samples.

The confidence interval is determined by the chosen confidence level (in the exercise, 95%), the sample size, the population standard deviation, and the Z-score corresponding to the confidence level.

The margin of error represents the extent to which the estimate obtained from the sample can differ from the true population parameter. In simpler terms, it’s an expression of the level of uncertainty or the maximum expected difference in the study.

A smaller margin of error indicates a more precise measurement and is determined mainly by the size of the sample and the standard deviation of the population. The formula for the margin of error is given by the expression \( E = Z \cdot \frac{\sigma}{\sqrt{n}} \), where \( E \) is the margin of error, \( Z \) is the Z-score, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. The exercise provided a target margin of error of 10, which is then used with a Z-score and the population standard deviation to calculate the required sample size.

The population standard deviation (\( \sigma \) ) is a measure of the dispersion or variability in a set of data. It quantifies how spread out the numbers are in the population.

For instance, if the population standard deviation is low, it indicates that the values are closely clustered around the mean. Conversely, a high standard deviation suggests that the values are spread out over a wide range.

In the context of sample size determination, the population standard deviation plays a crucial role: a larger standard deviation would typically require a larger sample size to achieve a specific margin of error for a confidence interval. In the exercise, a population standard deviation of 40 indicates variability in the data, which directly impacts the margin of error and, subsequently, the required sample size.

The 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. The standard normal distribution, which the Z-score is based on, allows us to determine how far away a particular point is from the population mean, expressed in standard deviations.

A Z-score of 1.96, as used in this exercise for a 95% confidence level, tells us that the value lies nearly two standard deviations from the mean. This Z-score is a crucial component in determining the confidence interval and the margin of error, as it encapsulates the desired level of confidence in the statistical conclusions. The Z-score contributes to calculating how large a sample size needs to be for the estimated mean to fall within a certain range of the true population mean.