In Problem 3, assume that the paired differences come from a population that is normally distributed. 
a. Compute di=Xi−Yi for each pair of data.  

Verified step by step guidance

1. Identify the paired data values for each observation from the table. For each observation i, you have two values: \(X_i\) and \(Y_i\).
2. Recall that the difference for each pair is calculated as \(d_i = X_i - Y_i\). This means you subtract the value of \(Y_i\) from the corresponding \(X_i\) for each observation.
3. Perform the subtraction for each pair: for observation 1, calculate \(d_1 = 34.2 - 34.9\); for observation 2, calculate \(d_2 = 32.1 - 31.5\); and so on for all six observations.
4. List all the differences \(d_i\) obtained from the calculations in a new column or list to keep track of the paired differences clearly.
5. Once all differences are computed, you can use these values for further analysis, such as calculating the mean difference, standard deviation, or conducting hypothesis tests assuming the differences come from a normally distributed population.

Key Concepts

Paired Differences

Paired differences are calculated by subtracting one value from another within matched pairs of data, typically denoted as di = Xi - Yi. This approach is used to analyze the difference between two related measurements for the same subject or entity, allowing for direct comparison.

Normal Distribution Assumption

Assuming the paired differences come from a normally distributed population means the differences follow a bell-shaped curve. This assumption is crucial for applying parametric tests like the paired t-test, which rely on normality to make valid inferences about the population.

Data Organization and Computation

Organizing data in a table format helps systematically compute paired differences for each observation. Careful calculation of di for each pair ensures accuracy in subsequent statistical analysis, such as hypothesis testing or confidence interval estimation.