Answer :
To determine the proportion of data falling between 85 and 100, we can use the 68-95-99.7 Rule in statistics. This rule applies to data that is normally distributed, meaning it follows a bell-shaped curve.
Let's break down the steps:
Understand the Mean and Standard Deviation: The mean of the data is given as 100, and the standard deviation is 5. This means that most of the data clusters around the mean, with a spread measured by the standard deviation.
Identify the 68-95-99.7 Rule:
- 68% of the data falls within one standard deviation of the mean (from [tex]\mu - \sigma[/tex] to [tex]\mu + \sigma[/tex]).
- 95% of the data falls within two standard deviations of the mean (from [tex]\mu - 2\sigma[/tex] to [tex]\mu + 2\sigma[/tex]).
- 99.7% of the data falls within three standard deviations of the mean (from [tex]\mu - 3\sigma[/tex] to [tex]\mu + 3\sigma[/tex]).
Calculate the Range for One Standard Deviation:
- One standard deviation below the mean: [tex]100 - 5 = 95[/tex]
- Therefore, 68% of the data is between 95 and 105.
Determine the Lower Range (85 to 100):
- Note that 85 is three standard deviations below the mean, as [tex]100 - 3\times5 = 85[/tex].
- To find the proportion between 85 and 100, consider 100 as the endpoint.
Distribution under the Bell Curve:
- From 85 to 100, we cover half of the data below the mean (as the entire left side of the normal distribution curve). This represents half of the 50% that lies below the mean.
- Therefore, 50% is below the mean. Since we are concerned with one side of this 50%, we look at the normal distribution graph: from mean to left is 50%.
- In total, looking at the symmetric properties of the normal distribution, approximately 50-44 = 6% lies below the mean but within the first standard deviation on one side i.e., between 95 and 100.
- Adding that 6% to 50% gives 56% for the range to be below mean between 95-100
Therefore, the proportion of the data that falls between 85 and 100 is about 56%.
