A certain standardized test’s math scores have a bell-shaped distribution with a mean of 525 and a standard deviation of 105. Complete parts
(a) through (c). (a) What percentage of standardized test scores is between 210 and 840?
99.7% (Round to one decimal place as needed.)
(b) What percentage of standardized test scores is less than 210 or greater than 340?
(Round to one decimal place as needed.)
The correct answer and explanation is :
(a) What percentage of standardized test scores is between 210 and 840?
To determine the percentage of standardized test scores between 210 and 840, we can use the properties of a bell-shaped (normal) distribution. The mean is 525, and the standard deviation is 105.
We can use the Empirical Rule, also known as the 68-95-99.7 rule, which states:
- 68% of the data falls within 1 standard deviation of the mean.
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of the data falls within 3 standard deviations of the mean.
To find how many standard deviations away 210 and 840 are from the mean:
- For 210:
[
Z = \frac{210 – 525}{105} = \frac{-315}{105} = -3
] - For 840:
[
Z = \frac{840 – 525}{105} = \frac{315}{105} = 3
]
Thus, 210 is 3 standard deviations below the mean, and 840 is 3 standard deviations above the mean.
According to the Empirical Rule, 99.7% of the data lies between 3 standard deviations below and 3 standard deviations above the mean. Therefore, the percentage of standardized test scores between 210 and 840 is 99.7%.
(b) What percentage of standardized test scores is less than 210 or greater than 840?
To find the percentage of scores less than 210 or greater than 840, we need to calculate the area outside the range of 210 and 840.
Since 99.7% of the data is between 210 and 840 (from part (a)), the remaining 0.3% of the data is outside this range. This remaining percentage is split equally into two tails of the distribution.
Thus, each tail has:
[
\frac{0.3\%}{2} = 0.15\%
]
This means 0.15% of the data is less than 210, and 0.15% of the data is greater than 840.
Therefore, the total percentage of scores less than 210 or greater than 840 is:
[
0.15\% + 0.15\% = 0.3\%
]
Explanation:
In a normal distribution, the data is symmetrically distributed around the mean. The Empirical Rule provides a quick way to estimate the percentage of data within a certain number of standard deviations from the mean. By converting the raw scores (210 and 840) into z-scores and using the rule, we can easily determine the percentage of data that falls within or outside a specified range.