Question

Exercises 8.41 to 8.45 refer to the data with analysis shown in the following computer output: \(\begin{array}{lrrr}\text { Level } & \text { N } & \text { Mean } & \text { StDev } \\ \text { A } & 5 & 10.200 & 2.864 \\ \text { B } & 5 & 16.800 & 2.168 \\ \text { C } & 5 & 10.800 & 2.387\end{array}\) \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Groups } & 2 & 133.20 & 66.60 & 10.74 & 0.002 \\ \text { Error } & 12 & 74.40 & 6.20 & & \\ \text { Total } & 14 & 207.60 & & & \end{array}\) Find a \(90 \%\) confidence interval for the difference in the means of populations \(\mathrm{B}\) and \(\mathrm{C}\).

Short answer

The 90% confidence interval for the difference in means of populations B and C can be calculated as \( 16.8 - 10.8 \pm 1.860*SE_d \), substituting the calculated SE

Step-by-step solution

Compute Standard Error

The Standard Error (SE) for the difference in means can be obtained using the formula: \( SE_d = \sqrt{(s_{1}^{2}/n_{1}) + (s_{2}^{2}/n_{2})} \). Substituting the given values, \( s_{1} = 2.168 \), \( s_{2} = 2.387 \), \( n_{1} = n_{2} = 5 \) into the formula, we get \( SE_d = \sqrt{(2.168^2/5) + (2.387^2/5)} \)

Compute degrees of freedom

Since we are using t-distribution, we need degrees of freedom (DF), which is \(DF=n_{1}+n_{2}-2 = 5+5-2 = 8\). Using this DF, we look at the t-distribution table to find the t-value that corresponds to the 90% confidence level, which is \(1.860\).

Calculate Confidence Interval

Using the formula for confidence interval for difference in means \(\mu_{1} - \mu_{2} \pm t* SE_d \), and substituting the calculated and given values, \( \mu_{1} = 16.8 \), \( \mu_{2} = 10.800 \), \( t=1.860 \), we get the confidence interval as \( 16.8 - 10.8 \pm 1.860*SE_d \).

Key concepts

Understanding Standard Error (SE)

When conducting statistics, it's essential to have an accurate measure of the variability or precision of a sample mean compared to the population mean. This is where Standard Error (SE) comes into play. The SE provides insight into how much a sample mean is expected to differ from the true population mean purely by chance.

Mathematically, the SE for the difference between two independent means is calculated as \( SE_d = \sqrt{(s_{1}^{2}/n_{1}) + (s_{2}^{2}/n_{2})} \), where \(s_{1}^{2}\) and \(s_{2}^{2}\) are the variances (squared standard deviations) of the two samples, and \(n_{1}\) and \(n_{2}\) are the sample sizes. Lower SE values signify greater precision in estimating the population mean, as they indicate less variability between the sample means and the true population mean.

For our problem, by substituting the given values into the formula, we can find the SE required to further calculate the confidence interval for the difference in means between populations B and C. A smaller SE potentially implies that the difference between the sample means is a more reliable estimator of the difference between the population means.

Interpreting the t-distribution

Working with Small Sample Sizes

When it comes to small sample sizes, normal distribution assumptions do not always hold. That's where the t-distribution comes in handy. It's a type of probability distribution that is symmetric and bell-shaped, much like the normal distribution, but with thicker tails, which means it accounts for more variability.

The t-distribution is particularly useful when dealing with estimates derived from small sample sizes (<30) or when the population standard deviation is unknown. With smaller samples, there is more uncertainty, which the t-distribution helps to account for. By using the t-distribution, we can generate more accurate confidence intervals and hypothesis tests for our sample data.

In our exercise, with degrees of freedom (DF) of 8, we use the t-distribution to find the appropriate t-value for constructing the confidence interval for the difference in means. The t-value, determined from statistical tables or software, corresponds to the desired level of confidence—in this case, 90%.

Degrees of Freedom Explained

The concept of degrees of freedom (DF) is a fundamental aspect of statistical analysis, which corresponds to the number of independent pieces of information we have while estimating a statistic. In the context of comparing two means, DF is calculated as \(DF = n_{1} + n_{2} - 2\), where \(n_{1}\) and \(n_{2}\) are the sample sizes for the respective groups.

Why Are Degrees of Freedom Important?

DF play a crucial role in distributions like the t-distribution that adjusts its shape based on the number of observations in the data. The more degrees of freedom we have, the closer the t-distribution gets to the normal distribution. For smaller samples, fewer degrees of freedom lead to wider confidence intervals, reflecting increased uncertainty.

In our example, with 5 observations in each group, we have 8 degrees of freedom (10 total observations minus the 2 means we are estimating). This value is required when we refer to the t-distribution to obtain the accurate critical value needed for constructing the confidence interval.