Question

Use the t-distribution to find a confidence interval for a difference in means \(\mu_{1}-\mu_{2}\) given the relevant sample results. 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. A \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the sample results \(\bar{x}_{1}=501, s_{1}=115, n_{1}=400\) and \(\bar{x}_{2}=469, s_{2}=96, n_{2}=200 .\)

Short answer

The best estimate for \( \mu_{1}-\mu_{2}\) is 32. The margin of error is approximately 19.31. A 99% confidence interval for \( \mu_{1}-\mu_{2}\) given the sample results is approximately [12.69, 51.31].

Step-by-step solution

Calculate the Difference in Sample Means

First, calculate the difference in sample means. This calculation gives us the best estimate for \(\mu_{1}-\mu_{2}\). The formula is \(\bar{x}_{1} - \(\bar{x}_{2}\). Substitute \(\bar{x}_{1}=501\) and \(\bar{x}_{2}=469\), yielding \(501 - 469 = 32\). So the best estimate for \(\mu_{1}-\mu_{2}\) is 32.

Calculate Standard Error

The next step involves calculating the standard error. Use the formula \(\sqrt{\frac{{s_1}^2}{{n_1}} + \frac{{s_2}^2}{{n_2}}}\). Substituting \(s_{1}=115, n_{1}=400, s_{2}=96, n_{2}=200\), this yields \(\sqrt{\frac{{115}^2}{400} + \frac{{96}^2}{200}}\), which is approximately 7.392.

Look up the t-value

A t-value is needed because the exercise instructs us to use the t-distribution. For a \(99%\) confidence interval and the given sample sizes, the degrees of freedom will be the smaller of \(n_{1}-1\) and \(n_{2}-1\), which is \(200 - 1 = 199\). Looking this up in a t-table, we find the t-value for a \(99%\) confidence level and \(199\) degrees of freedom is approximately \(2.61\).

Calculate Margin of Error

The margin of error is calculated by multiplying the t-value by the standard error. This yields \(2.61 * 7.392\), approximately \(19.31\).

Construct the Confidence Interval

Finally, the confidence interval is constructed by adding and subtracting the margin of error from the best estimate. So, the confidence interval is \([32 - 19.31, 32 + 19.31] = [12.69, 51.31]\).

Key concepts

t-distribution

The t-distribution, also known as Student's t-distribution, is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population's standard deviation is unknown. As the sample size increases, the t-distribution approaches the normal distribution.

In the context of confidence intervals for the difference in means, the t-distribution is used because it accommodates the extra variability due to estimating the population standard deviation from the sample. This is particularly important when dealing with small sample sizes or when the sample standard deviations are different. When constructing a confidence interval using the t-distribution, we must also consider the degrees of freedom, which impacts the width of the confidence interval.

Standard Error Calculation

The standard error measures the uncertainty associated with a sample estimate. For the difference between two sample means, the standard error reflects how much variability there is in estimating the difference between the two population means. The formula to calculate the standard error of the difference in means is \( \sqrt{\frac{{s_1}^2}{{n_1}} + \frac{{s_2}^2}{{n_2}}} \), where \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.

The standard error accounts for how the variance within each group and the size of each sample affect the overall variability and thus provides a way to quantify the precision of the estimated difference in means. The larger the standard error, the less certain we are about the estimate of the difference in means.

Degrees of Freedom

Degrees of freedom are a concept used in various statistical analyses, including t-distribution and determination of the margin of error for the difference in sample means. They can be thought of as the number of values in the final calculation of a statistic that are free to vary.

For the difference in means, the degrees of freedom are usually calculated based on the sample sizes of the two groups. In the t-distribution, they are crucial as they determine the specific shape of the distribution, which impacts the critical value or 't-value' used in confidence interval calculations. The degrees of freedom are generally calculated as the smaller of \(n_{1} - 1\) or \(n_{2} - 1\). A larger degree of freedom indicates a distribution more closely resembling a normal distribution and results in a narrower confidence interval, assuming a fixed confidence level.

Margin of Error

The margin of error represents the range of values above and below the sample estimate that you expect the true value to fall into. It reflects the maximum expected difference due to sampling variability. In the context of estimating the difference in means, it is calculated by multiplying the standard error by the t-value associated with the desired confidence level and degrees of freedom.

The formula is \( E = t \times SE \), where \( E \) is the margin of error, \( t \) is the t-value from the t-distribution, and \( SE \) is the standard error. The margin of error gives us a range centered around our point estimate (the difference in sample means), which constructs the confidence interval. It's a crucial component because it helps us understand the precision of our interval estimate and how much the true parameter value could plausibly vary from our best estimate.