Sure, let's break this down step-by-step.
Given:
- Mean (μ) = 84
- Standard Deviation (σ) = 10
### 1. 95% of the Data Values
According to the Empirical Rule (68-95-99.7 Rule):
- 95% of the data in a normal distribution lies within 2 standard deviations of the mean.
Calculation:
- Lower Bound: Mean - 2 × Standard Deviation = 84 - 2 × 10 = 64
- Upper Bound: Mean + 2 × Standard Deviation = 84 + 2 × 10 = 104
Thus, 95% of the data values lie between 64 and 104.
### 2. Percentage of Exam Scores Less Than or Equal to 84
By definition of the mean in a normal distribution:
- 50% of the exam scores are less than or equal to the mean value (84).
Thus, 50% of the exam scores are less than or equal to 84.
### 3. Percentage of Exam Scores Less Than or Equal to 74
To find this, we need to calculate the z-score for 74:
- Z = (X - μ) / σ = (74 - 84) / 10 = -1
Then we find the cumulative probability for z = -1.
From standard normal distribution tables or using a statistical tool, the cumulative probability for z = -1 is approximately 0.158655.
Thus, approximately 15.87% of the exam scores are less than or equal to 74.
### 4. Percentage of Exam Scores Less Than or Equal to 104
To find this, we need to calculate the z-score for 104:
- Z = (X - μ) / σ = (104 - 84) / 10 = 2
The cumulative probability for z = 2.
From standard normal distribution tables or using a statistical tool, the cumulative probability for z = 2 is approximately 0.97725.
Thus, approximately 97.72% of the exam scores are less than or equal to 104.
### 5. Percentage of Exam Scores Less Than or Equal to 94
To find this, we need to calculate the z-score for 94:
- Z = (X - μ) / σ = (94 - 84) / 10 = 1
The cumulative probability for z = 1.
From standard normal distribution tables or using a statistical tool, the cumulative probability for z = 1 is approximately 0.84134.
Thus, approximately 84.13% of the exam scores are less than or equal to 94.
### Final Summary:
1. 95% of the data values lie between 64 and 104.
2. 50% of the exam scores are less than or equal to 84.
3. Approximately 15.87% of the exam scores are less than or equal to 74.
4. Approximately 97.72% of the exam scores are less than or equal to 104.
5. Approximately 84.13% of the exam scores are less than or equal to 94.
