Problem 28 - An Introduction to Mathematical Statistics and Its Applications

Question

Two samples, each of size \(n\), are taken from a normal distribution with unknown mean \(\mu\) and unknown standard deviation \(\sigma .\) A \(90 \%\) confidence interval for \(\mu\) is constructed with the first sample, and a \(95 \%\) confidence interval for \(\mu\) is constructed with the second. Will the \(95 \%\) confidence interval necessarily be longer than the \(90 \%\) confidence interval? Explain.

Step-by-Step Solution

Answer

Yes, a 95% confidence interval will necessarily be longer than a 90% confidence interval. This is because a higher confidence level requires a wider interval to account for more possibilities.

Step 1: Defining Confidence Interval

Firstly, the concept of a confidence interval should be understood. A confidence interval is defined as a range of values that are likely to contain the true parameter with a certain degree of confidence. The wider the interval, the higher the level of confidence.

Step 2: Understanding Confidence Level

A 90% confidence interval means there is a 90% probability that the range contains the true parameter. Similarly, a 95% confidence interval implies a 95% probability that the range contains the true parameter. Thus, the width of the confidence interval increases with increasing the level of confidence.

Step 3: Comparing Confidence Intervals

As the confidence level rises from 90% to 95%, the range of possible values that could contain the true parameter also increases. Thus, a 95% confidence interval is typically larger than a 90% confidence interval.

Key Concepts

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics and probability. It is characterized by its bell-shaped curve, which is symmetric around the mean. The mean, median, and mode are all equal in a normal distribution. This makes it particularly useful for understanding the distribution of a dataset because many variables tend to follow a normal pattern.

Key features include:

• The curve is symmetric about the center.
• The spread of the curve is determined by the standard deviation.
• Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Importantly, when we assume a normal distribution, it simplifies the process of creating confidence intervals which helps in parameter estimation.

Standard Deviation

Standard deviation is a measure of how spread out numbers are around the mean. In the context of a normal distribution, it gives us insight into the variability or dispersion of the dataset. A small standard deviation indicates that the values tend to be close to the mean, while a large standard deviation indicates that the values are spread out over a wider range.

This is crucial when constructing confidence intervals. The standard deviation impacts the width of the interval, allowing us to estimate the range in which a true parameter should lie with a certain confidence level. Understanding this measure helps in interpreting the normal distribution accurately.

Confidence Level

The confidence level in a statistical analysis is the probability that a certain parameter lies within the confidence interval. For example, a 90% confidence level implies there's a 90% chance that the true parameter is contained within the interval computed from your sample.

Higher confidence levels result in wider intervals because they account for more potential outcomes. This means a 95% confidence interval is usually broader than a 90% one, given the same sample and standard deviation. This concept explains why, with more confidence, we need a wider range to ensure we capture the true parameter.

The selection of a confidence level depends on how much certainty you need for your estimations.

Parameter Estimation

Parameter estimation refers to the process of using sample data to estimate the parameters of a distribution, such as the mean or standard deviation. It is a vital aspect of statistics that aids in making inferences about a population.

Using confidence intervals is one method for parameter estimation. By calculating a range of values, we can estimate where the true parameter value lies. This method considers both the sample data and the confidence level to provide a reliable estimate.

In practical terms, good parameter estimation allows researchers to make informed decisions based on the analysis of their data.