A certain standardized test’s math scores have a bell-shaped distribution with a mean of 530 and a standard deviation of 105. Complete parts (a) through (c).
(a) What percentage of standardized test scores is between 425 and 635?
% (Round to one decimal place as needed.)
(b) What percentage of standardized test scores is less than 425 or greater than 635?
% (Round to one decimal place as needed.)
(c) What percentage of standardized test scores is greater than 740?
% (Round to one decimal place as needed.)
The correct answer and explanation is :
To answer these questions, we’ll use the properties of the normal distribution. A bell-shaped distribution means the data follows a normal distribution, with the mean and standard deviation provided.
- Mean (μ) = 530
- Standard deviation (σ) = 105
(a) What percentage of standardized test scores is between 425 and 635?
First, we calculate how many standard deviations away 425 and 635 are from the mean using the Z-score formula:
[
Z = \frac{X – \mu}{\sigma}
]
Where:
- (X) is the raw score (either 425 or 635),
- (\mu) is the mean (530),
- (\sigma) is the standard deviation (105).
For 425:
[
Z_{425} = \frac{425 – 530}{105} = \frac{-105}{105} = -1
]
For 635:
[
Z_{635} = \frac{635 – 530}{105} = \frac{105}{105} = 1
]
Now, we need to find the percentage of scores between Z = -1 and Z = 1. This corresponds to the area under the normal curve between these two Z-scores. Using standard Z-tables or a calculator, we find that the area between Z = -1 and Z = 1 is approximately 68%. Therefore, 68% of the scores fall between 425 and 635.
(b) What percentage of standardized test scores is less than 425 or greater than 635?
This is asking for the percentage of scores outside the range between 425 and 635. From part (a), we know that 68% of the scores are between these values. Since the total percentage of data in a normal distribution is 100%, the percentage outside this range is:
[
100\% – 68\% = 32\%
]
Thus, 32% of the scores are either less than 425 or greater than 635.
(c) What percentage of standardized test scores is greater than 740?
Now, we need to calculate how many standard deviations 740 is above the mean:
[
Z_{740} = \frac{740 – 530}{105} = \frac{210}{105} = 2
]
The Z-score of 2 corresponds to the area to the left of Z = 2 in a standard normal distribution. From Z-tables or a calculator, the area to the left of Z = 2 is approximately 0.9772, meaning that 97.72% of the data is less than 740. Therefore, the percentage greater than 740 is:
[
100\% – 97.72\% = 2.28\%
]
So, 2.3% of the standardized test scores are greater than 740.
Summary of Answers:
- (a) 68.0% of the scores are between 425 and 635.
- (b) 32.0% of the scores are less than 425 or greater than 635.
- (c) 2.3% of the scores are greater than 740.