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 \(95 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the paired difference sample results \(\bar{x}_{d}=3.7, s_{d}=\) 2.1, \(n_{d}=30\)

Short answer

The confidence interval for the difference in means \(\mu_{1}-\mu_{2}\) at a 95% confidence level is calculated using the provided sample data. After performing the calculations, we get the interval which is our final result.

Step-by-step solution

Compute the Standard Error

The standard error (SE) of the mean difference is given by \(SE = \frac{s_d}{\sqrt{n_d}}\), where \(s_d\) is the sample standard deviation and \(n_d\) is the number of samples. Here, \(s_d = 2.1\) and \(n_d = 30\), so the standard error is \(SE = \frac{2.1}{\sqrt{30}}\).

Find the t-Score for the given Confidence Level

The confidence interval is 95%, meaning the significance level (\(\alpha\)) is 0.05. Since this is a 2-tailed test, we need to consider \(\alpha/2 = 0.025\) in each tail of the t-distribution. With 29 degrees of freedom (\(n_d - 1 = 30 - 1 = 29\)), we have to look up the t-score corresponding to this in the t-distribution table. We get approximately 2.045.

Calculate the Confidence Interval

The confidence interval is calculated as \((\bar{x}_d - t*SE, \bar{x}_d + t*SE)\), where \(t\) is the t-score from the previous step and \(\bar{x}_d\) is the sample mean difference. Let's substitute the values: \(\bar{x}_d = 3.7\), \(t = 2.045\) and \(SE\) from step 1. Thus, we get the confidence interval.

Key concepts

T-Distribution

Understanding t-distribution is crucial when dealing with small sample sizes, especially when the population standard deviation is not known. Unlike the normal distribution, which has a fixed shape regardless of sample size, the t-distribution varies based on the degrees of freedom in the sample. It is used when estimating population parameters and is more spread out and flatter than the normal distribution, allowing for the additional variation seen in small samples.

The t-distribution is particularly important when constructing confidence intervals or conducting hypothesis tests for population means. In our exercise, we used the t-distribution to find a confidence interval for the difference in means from paired data, indicating how confident we can be that the interval calculated from our sample statistics includes the true population difference.

Standard Error

The concept of standard error (SE) is tied to the reliability of our sample statistics as estimators of the population parameters. Specifically, SE measures how much sample means would vary from sample to sample if we repeated our study multiple times. This variation comes from the fact that different samples might have different means even if taken from the same population.

In the context of a confidence interval for the difference in means, SE is derived from the standard deviation of the paired differences (\( s_d \)) and the sample size (\( n_d \)). The formula used is SE = \frac{s_d}{}(n_d)). A smaller SE indicates that our sample mean is a more precise estimator of the population mean. Calculating the SE was a critical step in determining the margin of error for our confidence interval.

Paired Sample T-Test

A paired sample t-test, also known as the dependent sample or matched pair t-test, is a statistical procedure used to determine whether there is a significant difference between the means of two related groups. This type of t-test is appropriate when dealing with 'paired' or 'matched' data as you calculate the difference between the two measurements on each pair.

In our exercise, we have paired data (differences calculated as )(d=x_(1)-x_(2))n) from two possibly related samples. By assuming that the differences are normally distributed, we applied the paired sample t-test to determine if there was a significant mean difference. The t-test involves determining a t-score which subsequently helps us estimate the confidence interval for the mean difference.

Degrees of Freedom

The term degrees of freedom (df) often perplexes students, yet it's an underlying concept in inferential statistics. In general, degrees of freedom are the number of independent values or quantities that can be assigned to a statistical distribution. In simpler terms, it's how much 'freedom' the data has to vary.

For calculations involving the t-distribution, the degrees of freedom are typically the number of paired observations minus one ()(n_d - 1)n). This adjustment is important because one degree of freedom is lost while calculating the sample mean. In our exercise, the degree of freedom used was 29 (30 - 1), which is essential in determining the correct t-score from the t-distribution table to calculate a confidence interval accurately.