The College Board, which are administered each year to many thousands of high school students, are scored so as to yield a mean of 550 and a standard deviation of 100. these scores are close to being normally distributed. what percentage of the scores can be expected to satisfy each condition?
Can you check these answers first?
greater than 600 =30.853
greater than 750=2.275
less than 400=6.681
between 400 and 600=62.465
what score represents Q3?
what score separates the top 70% from the bottom 30% ?
To check the given answers, we can calculate the percentages using the standard normal distribution table or a calculator.
To start, let's calculate the z-scores for each given value and find the corresponding percentage.
1. Greater than 600:
First, we need to calculate the z-score for 600:
z = (x - mean) / standard deviation
z = (600 - 550) / 100 = 0.5
Using the standard normal distribution table, the percentage for z > 0.5 is approximately 0.3085, or 30.85%. This matches the given answer of 30.853%.
2. Greater than 750:
Similarly, we find the z-score for 750:
z = (750 - 550) / 100 = 2
The percentage for z > 2 is approximately 0.0228, or 2.28%. This matches the given answer of 2.275%.
3. Less than 400:
The z-score for 400 is:
z = (400 - 550) / 100 = -1.5
The percentage for z < -1.5 is approximately 0.0668, or 6.68%. This is slightly different from the given answer of 6.681%.
4. Between 400 and 600:
For this range, we need to calculate the percentage between the two respective z-scores.
The z-score for 400: z = (400 - 550) / 100 = -1.5
The z-score for 600: z = (600 - 550) / 100 = 0.5
Using the standard normal distribution table, the percentage between -1.5 and 0.5 is approximately 0.6247, or 62.47%. This matches the given answer of 62.465%.
Now let's answer the additional questions:
5. What score represents Q3 (the third quartile)?
To find Q3, we need to determine the z-score that corresponds to the cumulative percentage of 0.75 (the third quartile).
Using the standard normal distribution table, the z-score for a cumulative percentage of 0.75 is approximately 0.675.
Now, we can relate the z-score to the actual score:
z = (x - mean) / standard deviation
0.675 = (x - 550) / 100
0.675 * 100 = x - 550
67.5 + 550 = x
x = 617.5
Therefore, the score that represents Q3 is approximately 617.5.
6. What score separates the top 70% from the bottom 30%?
To find the score that separates the top 70% from the bottom 30%, we need to determine the z-score that corresponds to the cumulative percentage of 0.30.
Using the standard normal distribution table, the z-score for a cumulative percentage of 0.30 is approximately -0.524.
Now, we can relate the z-score to the actual score:
-0.524 = (x - 550) / 100
-0.524 * 100 = x - 550
-52.4 + 550 = x
x = 497.6
Therefore, the score that separates the top 70% from the bottom 30% is approximately 497.6.
I hope this explanation helps you understand how to calculate the percentages and solve similar problems in the future!