Answer :
B. 68% of the data falls within one standard deviation of the mean in a normal distribution.
To find the percentage of scores within the range of 2.29 and 3.71 based on the differences calculated from the study, we follow these steps:
Differences in perceived quality are
3, 4, 3, 3, 2.
Calculate the mean:
(3 + 4 + 3 + 3 + 2) / 5 = 15 / 5 = 3.
Calculate the standard deviation:
First, find the variance:
((3 - 3)² + (4 - 3)² + (3 - 3)² + (3 - 3)² + (2 - 3)²) / 5
= (0 + 1 + 0 + 0 + 1) / 5 = 2 / 5 = 0.4.
The standard deviation is the square root of the variance:
√0.4 ≈ 0.632.
Calculate the range within 2 standard deviations:
Lower limit: Mean - 2 × Standard Deviation
= 3 - 2 × 0.632 ≈ 1.736.
Upper limit: Mean + 2 × Standard Deviation
= 3 + 2 × 0.632 ≈ 4.264.
Since the values 2.29 and 3.71 fall within the calculated range of approximately 1.736 to 4.264, we can use the empirical rule. This rule states that approximately 68% of the data falls within one standard deviation of the mean in a normal distribution.
Therefore, the correct answer is (B) 68%.