Sure, let's answer each part of the question step by step.
1. 95% of the data values lie between:
By the Empirical Rule, approximately 95% of the data values in a normally distributed set lie within two standard deviations of the mean.
- The mean of the exam scores is 84.
- The standard deviation is 10.
Therefore, we calculate the range as follows:
- 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 scores less than or equal to 84:
Since the mean is the middle value in a normal distribution, 50% of the exam scores are less than or equal to the mean (which is 84).
Therefore, 50% of the exam scores are less than or equal to 84.
3. Percentage of scores less than or equal to 74:
To find the percentage of exam scores less than or equal to 74, we use the cumulative distribution function (CDF) of the normal distribution.
Given that the mean is 84 and the standard deviation is 10, the percentage of scores ≤ 74 is approximately 15.87%.
4. Percentage of scores less than or equal to 104:
To find the percentage of exam scores less than or equal to 104, we use the CDF of the normal distribution.
Given that the mean is 84 and the standard deviation is 10, the percentage of scores ≤ 104 is approximately 97.72%.
5. Percentage of scores less than or equal to 94:
To find the percentage of exam scores less than or equal to 94, we also use the CDF of the normal distribution.
Given that the mean is 84 and the standard deviation is 10, the percentage of scores ≤ 94 is approximately 84.13%.
Summarizing the answers:
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. 15.87% of the exam scores are less than or equal to 74.
4. 97.72% of the exam scores are less than or equal to 104.
5. 84.13% of the exam scores are less than or equal to 94.
