Question

Use a t-distribution to find a confidence interval for the difference in means \(\mu_{1}-\mu_{2}\) using the relevant sample results from paired data. Give the best estimate for \(\mu_{1}-\) \(\mu_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples from populations that are approximately normally distributed, and that differences are computed using \(d=x_{1}-x_{2}\). A \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired difference sample results \(\bar{x}_{d}=556.9, s_{d}=\) \(143.6, n_{d}=100\)

Short answer

The best estimate for the difference of means ( \(\mu_{1}-\mu_{2}\) ) is 556.9, the margin of error is 27.8936, and the 90% confidence interval for the difference in means is (529.0064, 584.7936)

Step-by-step solution

Identify given values

From the problem, the mean difference ( \(\bar{x}_{d}\) ) is 556.9, the standard deviation of the differences ( \(s_{d}\)) is 143.6, and the sample size ( \(n_{d}\) ) is 100. The confidence level is 90%.

Calculate the t-value

We find the t-value that corresponds to our desired confidence level and degrees of freedom. Degrees of freedom, in this case, equals \( n_{d} - 1\), which is 99. For a 90% confidence level, the t-value (two-tailed) for 99 degrees of freedom is approximately 1.660.

Calculate the margin of error

The formula for the margin of error \( E \) with a t-distribution is \( E = t \cdot \frac{s_{d}}{\sqrt{n_{d}}} \). Substituting with provided values, \( E = 1.660 \cdot \frac{143.6}{\sqrt{100}} \) which results in 27.8936.

Determine the confidence interval

Our confidence interval is determined as \( \bar{x}_d ± E \), thus our interval is \( 556.9 ± 27.8936 \)

Key concepts

Understanding t-distribution

When diving into statistics, the concept of the t-distribution is essential, especially when working with smaller sample sizes or when the population standard deviation is unknown. The t-distribution, or Student's t-distribution, is a type of probability distribution that is symmetric and bell-shaped, much like the standard normal distribution, yet with heavier tails.

Heavier tails indicate a higher probability of values further from the mean, which is crucial when dealing with uncertainty in estimates from smaller samples. As the sample size increases, the t-distribution approaches the normal distribution. The t-distribution comes into play when calculating confidence intervals or conducting hypothesis testing for the mean population value when the sample size is below 30 or the population variance is unknown. In such scenarios, the t-distribution provides more reliable and adjusted intervals and test results than the normal distribution.

Paired Sample t-Test Explained

The paired sample t-test, also known as the dependent t-test, is a powerful statistical procedure used to compare the means from two related groups. These groups are 'paired' because they are somehow related or matched.

Examples of such pairings include before-and-after observations on the same subjects, or matched subjects in different conditions. In the paired sample t-test, we calculate the difference \(d=x_{1}-x_{2}\) for each matched pair of observations and then analyze these differences using the t-test. This method helps to control for variability between subjects, as the comparison is based only on the changes within each pair, not between different pairs. As a result, this test can be more sensitive in detecting differences than two independent samples t-tests.

Calculating the Margin of Error

The concept of margin of error plays a pivotal role in understanding the precision of an estimate. It is the range above and below the observed statistic value that is believed to contain, with a certain level of confidence, the true population parameter.

Mathematically, the margin of error (\(E\)) can be calculated in context of a t-distribution as \( E = t \cdot \frac{s_{d}}{\sqrt{n_{d}}} \), where \(t\) is the t-distribution critical value associated with our desired confidence level, \(s_d\) is the standard deviation of the differences observed in our paired sample, and \(n_d\) is the number of differences or pairings. Higher confidence levels or larger variabilities result in a wider margin of error, suggesting less precision, while a larger sample size reduces the margin of error, indicating a more precise estimate.