From the Empirical Rule, we can deduce the following (select ALL that apply): (Note: IQR stands for the Interquartile Range and SD stands for Standard Deviation) Group of answer choices 1. The IQR is always larger than the SD. 2. The IQR is usually larger than the SD for bell-shaped distributions. 3. The SD is always larger than the IQR. 4. The SD is usually larger than the IQR for bell-shaped distributions. 5. The SD is equal to the IQR for bell-shaped distributions. 6. The SD is always equal to the IQR. b. Assume that the GMAT score follows a Normal Distribution with a mean of 569 and a population standard deviation of 73. Use ONLY the Empirical Rule (68-95-99.7 rule) to answer the following question: What is the GMAT score corresponding to the 16th percentile? Report your answer as a whole number. For example, if your answer is 575.89, report 576. c. Use ONLY the 68-95-99.7 (i.e. Empirical) rule to answer the following question. A 100-point homework assignment was given. David’s score on the homework assignment was at the 97.5th percentile. All scores for the homework assignment were normally distributed with a mean of 77 and a standard deviation of 5. David’s friend Brent received a score of 72 on his homework assignment. What proportion of students scored better than Brent? (Report value as a proportion NOT a percent and round to 2 decimal places.) d. Use ONLY the Empirical (68-95-99.7) Rule to answer the following question. A 100-point homework assignment was given. David’s score on the homework assignment was at the 97.5th percentile. Assuming that all scores for the homework assignment were normally distributed with a mean of 70 and a standard deviation of 6, what was David’s homework assignment score? Report your answer as a whole number (i.e. if your answer is 80.78%, report 81)
The Correct Answer and Explanation is:
a. From the Empirical Rule, select ALL that apply:
Correct answer(s):
4. The SD is usually larger than the IQR for bell-shaped distributions.
Explanation:
The Empirical Rule applies specifically to bell-shaped (normal) distributions. It tells us that:
- Approximately 68% of data falls within 1 SD of the mean.
- About 95% lies within 2 SDs.
- About 99.7% lies within 3 SDs.
The IQR (Interquartile Range) is the range between the 75th and 25th percentiles. In a normal distribution, the IQR captures the middle 50% of the data, and it’s approximately equal to 1.35 SD. Since 2 SDs span 95% of the data, this shows that SDs cover a broader range of data than the IQR. Therefore, for bell-shaped distributions, the standard deviation is usually larger than the IQR.
The other options incorrectly state fixed relationships or are untrue for bell-shaped distributions.
b. GMAT score at the 16th percentile (Mean = 569, SD = 73):
Using the Empirical Rule, the 16th percentile is approximately 1 standard deviation below the mean. Score=569−73=496\text{Score} = 569 – 73 = \boxed{496}Score=569−73=496
c. Proportion of students who scored better than Brent (Score = 72, Mean = 77, SD = 5):
Brent scored 1 standard deviation below the mean: Z=72−775=−1Z = \frac{72 – 77}{5} = -1Z=572−77=−1
According to the Empirical Rule:
- 68% of data falls within ±1 SD
- So, 34% fall between the mean and 1 SD below
- Therefore, below Brent (more than 1 SD below mean): 16%
- Thus, proportion who scored better = 1 – 0.16 = \boxed{0.84}
d. David’s score at the 97.5th percentile (Mean = 70, SD = 6):
The 97.5th percentile lies 2 SDs above the mean using the Empirical Rule: Score=70+(2×6)=82\text{Score} = 70 + (2 \times 6) = \boxed{82}Score=70+(2×6)=82
Summary:
- a) Answer: 4
- b) Answer: 496
- c) Answer: 0.84
- d) Answer: 82
