Problem 2 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

You are given a dataset that may be considered a realization of a normal random sample. The size of the dataset is 34 , the average is \(3.54\), and the sample standard deviation is \(0.13\). Construct a \(98 \%\) confidence interval for the unknown expectation \(\mu\).

Step-by-Step Solution

Answer

The 98% confidence interval for \( \mu \) is \( (3.488, 3.592) \).

Step 1: Identify the Components

For a confidence interval of a normal distribution, we need the sample size \( n = 34 \), sample mean \( \bar{x} = 3.54 \), and the sample standard deviation \( s = 0.13 \). Additionally, we are given a confidence level of \( 98\% \).

Step 2: Determine the Critical Value

Since the sample size is less than 30, we typically use a t-distribution, but here since the sample size is greater than 30, we will use a z-distribution. For a \( 98\% \) confidence level, the critical z-value (\( z^* \)) is approximately \( 2.33 \).

Step 3: Calculate the Standard Error

The standard error (SE) is calculated using the formula \( \text{SE} = \frac{s}{\sqrt{n}} \). Substituting the values, we get \( \text{SE} = \frac{0.13}{\sqrt{34}} \approx 0.02232 \).

Step 4: Compute the Margin of Error

Using the formula for the margin of error (ME) \( \text{ME} = z^* \times \text{SE} \), substitute the values: \( \text{ME} = 2.33 \times 0.02232 \approx 0.052 \).

Step 5: Construct the Confidence Interval

The confidence interval is given by \( (\bar{x} - \text{ME}, \bar{x} + \text{ME}) \). Substituting the known values, we find \( (3.54 - 0.052, 3.54 + 0.052) = (3.488, 3.592) \).

Key Concepts

Normal Distribution

The normal distribution is one of the most common probability distributions used in statistics, often called the bell curve due to its shape. It is symmetric around the mean, which means the distribution of data is uniform on both sides. In a normal distribution:

• The mean (\(\mu\)) is the center of the distribution.
• The standard deviation (\(\sigma\)) measures the spread of the data.
• About 68% of data falls within one standard deviation, 95% within two, and 99.7% within three.

Normal distributions are essential in the field of statistics as they describe many natural phenomena. However, they are just a model and real-world data can deviate from this idealized scenario. Our dataset is an example of a normal random sample, where we aim to make inferences about the population mean (\(\mu\)) based on the sample data provided.

Sample Standard Deviation

The sample standard deviation (\(s\)) is a measure of how spread out the numbers in a sample are. It is crucial for understanding the variability within a dataset:

• The formula for calculating the sample standard deviation is: \[s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}\]
• It is the square root of the sample variance.
• This standard deviation uses "n-1" in its denominator, known as Bessel's correction, to provide an unbiased estimation for population variance.

In our scenario, \(s = 0.13\), representing the average distance of each data point in our sample from the sample mean (\(\bar{x} = 3.54\)). This value plays a vital role in constructing confidence intervals as it affects the margin of error.

Z-distribution

The z-distribution, or standard normal distribution, is another fundamental concept in statistics. It refers to a normal distribution with a mean of 0 and a standard deviation of 1. The z-distribution is used extensively in hypothesis testing and confidence interval estimation:

• A z-score tells us how many standard deviations an element is from the mean.
• The critical value, \(z^*\), from the z-distribution is used to determine the specific confidence level, such as 95% or 98%.
• It becomes relevant when the sample size is relatively large, and the population standard deviation is known or approximated by the sample standard deviation.

In our exercise, we use \(z^* = 2.33\), which is the critical value for a 98% confidence level. This value helps in calculating the margin of error, which is pivotal in defining the confidence interval.

Margin of Error

The margin of error (\(\text{ME}\)) illustrates the range of uncertainty associated with a sample estimate when predicting population parameters. It is a critical component in confidence intervals and can be calculated using the critical value and standard error:

• The formula for the margin of error is: \[\text{ME} = z^* \times \text{SE}\]
• The standard error (\(\text{SE}\)) is derived from dividing the sample standard deviation by the square root of the sample size: \[\text{SE} = \frac{s}{\sqrt{n}}\]
• The resulting margin of error helps create a range around the sample mean where the population mean is likely to fall.

In our confidence interval calculation, \(\text{ME} \approx 0.052\). This margin reflects the amount we extend on either side of the sample mean (\(3.54\)) to express the confidence interval: \((3.488, 3.592)\). This interval provides not just a point estimate but a range that covers \(98\%\) likelihood of capturing the true population mean.