Question

Refer to Exercise \(11.46 .\) The means of two of the factor-level combinations- say, \(\mathrm{A}_{1} \mathrm{~B}_{1}\) and \(\mathrm{A}_{2} \mathrm{~B}_{1}-\) are \(\bar{x}_{1}=8.3\) and \(\bar{x}_{2}=6.3,\) respectively. Find a \(95 \%\) confidence interval for the difference between the two corresponding population means.

Short answer

Answer: The missing information needed to calculate the 95% confidence interval for the difference between two population means is the sample sizes and sample standard deviations for both samples.

Step-by-step solution

1. Identify the given information

We have the following information: - Sample means: \(\bar{x}_1 = 8.3\) and \(\bar{x}_2 = 6.3\) - Confidence level: 95%

2. Calculate the standard error

To compute the confidence interval, first, we need to calculate the standard error for the difference between the two population means. However, we don't have enough information to proceed with this step, as we need the sample sizes and sample standard deviations for both samples.

3. Determine the degree of freedom

As we don't have the sample sizes and standard deviations, we cannot compute the degrees of freedom either. The degrees of freedom are used to choose the appropriate t-value from the t-distribution table.

4. Calculate the confidence interval

Without the standard error and the appropriate t-value, we are unable to proceed with the calculation of the confidence interval for the difference between the two population means. We need more information about the samples to complete this exercise. If we had the required information, we would use the following formula to calculate the confidence interval: \(CI = (\bar{x}_1 - \bar{x}_2) \pm t \times SE\) Where \(CI\) denotes the confidence interval, \(\bar{x}_1\) and \(\bar{x}_2\) represent the sample means, \(t\) is the t-value from the t-distribution table, and \(SE\) stands for the standard error.

Key concepts

Sample Mean

The sample mean is a crucial element when calculating confidence intervals. In statistics, the sample mean represents the average of a set of data points collected from a larger population. It's denoted by \( \bar{x} \).

Imagine you want to guess the average height of students in a school, but instead of measuring everyone, you measure just a small group. The average height of this group is the sample mean.

When dealing with confidence intervals, we often compare multiple sample means to understand differences between groups. For instance, in our example above, we have \( \bar{x}_1 = 8.3 \) and \( \bar{x}_2 = 6.3 \), representing different group averages. These values help us estimate the difference between the population means they represent.

Standard Error

The standard error is a measure of the variability or spread of the sample mean. It helps determine how much the sample mean may deviate from the actual population mean. The smaller the standard error, the closer the sample mean is likely to be to the population mean.

To compute the standard error, you generally need the sample standard deviation and the sample size. The formula for a single sample's standard error is \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the sample standard deviation and \( n \) is the sample size.

In our initial formula for the confidence interval, the standard error helps us assess the "buffer" or "margin of error" around the sample mean difference \( \bar{x}_1 - \bar{x}_2 \). This margin assists in constructing the interval within which we expect the true population mean difference to occur.

t-Distribution

The t-distribution is a probability distribution used when estimating population parameters when the sample size is small and/or the population standard deviation is unknown. It's similar to the normal distribution but accounts for additional uncertainty by having "fatter" tails.

When calculating a confidence interval, especially with small samples or unknown population parameters, the t-distribution provides a critical value, denoted \( t \), to scale the standard error. This scaling adjusts the confidence interval to be more accurate given the circumstances.

The appropriate \( t \)-value depends on the desired confidence level (e.g., 95%) and the degrees of freedom, which usually relate to the sample size. You'd use a t-distribution table or statistical software to find this value.

While our exercise couldn't continue without specific sample sizes or standard deviations, understanding the t-distribution is vital for handling real-world data that isn't perfect.

Population Means

Population means are the average values we would obtain if we could measure every individual in the entire population. They're denoted using Greek symbols such as \( \mu \).

In statistics, we often can't measure the whole population, so we use sample means to estimate these population means. The goal of calculating confidence intervals is to find a range that likely contains the true population mean based on our sample data.

In comparing two population means, as per our exercise, we look at the difference between these means to understand if one group tends to have higher or lower values than another.

When constructing a confidence interval for a difference such as \( \mu_1 - \mu_2 \), our aim is to see where this difference likely exists based on sample observations, given the chosen confidence level like 95%. This approach helps maneuver through assumptions and uncertainties inherent in sampling.