Question

one hundred teachers attended a seminar on mathematical problem solving. the attitudes of a representative sample of 12 of the teachers were measured before and after the seminar. the differences between their pre-seminar scores and post-seminar scores are shown below.
10 6 0 -2 -2 2 1 0 2 -3 -2 1
(a) what is the range of the scores?
(b) what is the standard deviation of the scores? round your answer to the nearest tenth, if necessary.
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Explanation:

Step1: Find range (max - min)

First, identify the maximum value ($10$) and minimum value ($-3$) in the dataset.
Range = $10 - (-3) = 13$

Step2: Calculate sample mean

Sum all the values:
$10 + 6 + 0 + (-2) + (-2) + 2 + 1 + 0 + 2 + (-3) + (-2) + 1 = 13$
Mean $\bar{x} = \frac{13}{12} \approx 1.0833$

Step3: Compute squared deviations

For each value $x_i$, calculate $(x_i - \bar{x})^2$:
$(10-1.0833)^2 \approx 79.506$, $(6-1.0833)^2 \approx 24.173$, $(0-1.0833)^2 \approx 1.173$,
$(-2-1.0833)^2 \approx 9.506$, $(-2-1.0833)^2 \approx 9.506$, $(2-1.0833)^2 \approx 0.840$,
$(1-1.0833)^2 \approx 0.007$, $(0-1.0833)^2 \approx 1.173$, $(2-1.0833)^2 \approx 0.840$,
$(-3-1.0833)^2 \approx 16.673$, $(-2-1.0833)^2 \approx 9.506$, $(1-1.0833)^2 \approx 0.007$

Step4: Sum squared deviations & find variance

Sum of squared deviations:
$79.506 + 24.173 + 1.173 + 9.506 + 9.506 + 0.840 + 0.007 + 1.173 + 0.840 + 16.673 + 9.506 + 0.007 \approx 152.913$
Sample variance $s^2 = \frac{152.913}{12-1} \approx 13.901$

Step5: Calculate standard deviation

Sample standard deviation $s = \sqrt{13.901} \approx 3.7$

Answer:

(a) 13
(b) 3.7