Questions: Using the matched-pairs data below, calculate sd where di = Yi - Xi. Xi 14.36 15.63 18.80 16.95 19.52 16.41 14.24 14.59 Yi 17.47 17.92 16.86 16.47 17.37 18.58 20.14 17.34 Round your answer to the nearest thousandths. Answer =

Solution Steps

To calculate \( s_d \), the standard deviation of the differences \( d_i = Y_i - X_i \), follow these steps:

1. Compute the differences \( d_i \) for each pair of \( Y_i \) and \( X_i \).
2. Calculate the mean of these differences.
3. Compute the squared differences from the mean for each \( d_i \).
4. Find the variance by averaging these squared differences.
5. Take the square root of the variance to get the standard deviation \( s_d \).

For each pair of values \( (X_i, Y_i) \), calculate the difference \( d_i = Y_i - X_i \). The differences are: \[ d = [3.11, 2.29, -1.94, -0.48, -2.15, 2.17, 5.90, 2.75] \]

Calculate the mean of the differences: \[ \bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i = 1.4562 \]

For each difference \( d_i \), compute the squared difference from the mean: \[ (d_i - \bar{d})^2 = [2.7349, 0.6951, 11.5345, 3.7491, 13.0050, 0.5094, 19.7469, 1.6738] \]

Calculate the variance of the differences: \[ s_d^2 = \frac{1}{n} \sum_{i=1}^{n} (d_i - \bar{d})^2 = 6.7061 \]

Take the square root of the variance to find the standard deviation: \[ s_d = \sqrt{6.7061} = 2.5896 \]

Final Answer