College

Unit 1 Progress Check MCQ

Questions 13 through 15 refer to the following study:

A study was conducted to determine the power of a new chemical to increase the quality of olfactory sensations in humans. Participants were asked to rate the quality of a chocolate bar before and after inhaling the chemical on a scale of 1 to 10, with 1 meaning low quality and 10 meaning high quality. Analysis of the data showed that the difference in perceived quality of the chocolate bar before and after exposure to the chemical was statistically significant.

Perceived Quality of Chocolate Bar

[tex]
\[
\begin{array}{|c|c|c|c|}
\hline
& \text{BEFORE EXPOSURE} & \text{AFTER EXPOSURE TO CHEMICAL} & \text{DIFFERENCE (After - Before)} \\
\hline
1 & 5 & 8 & 3 \\
\hline
2 & 6 & 10 & 4 \\
\hline
3 & 3 & 6 & 3 \\
\hline
4 & 6 & 9 & 3 \\
\hline
5 & 5 & 7 & 2 \\
\hline
\end{array}
\]
[/tex]

Question 13

If these data were plotted on a graph, what percent of the scores for the perceived quality of the chocolate bar are within 2.29 and 3.71?

(A) 50
(B) 68
(C) 95
(D) 99.7

Answer :

Answer: 68%

Step-by-step explanation:

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%.

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