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% ?
Z = (score-mean)/SD
Z = (600-550)/100 = 50/100 = .5 = 30.85%
>750 = 2.28%
< 400 = 6.68%
>440 and < 600 = 62.47
Apparently your table has 5 decimal places.
Use the table to find the Z scores for .75 and .30 and insert those in the equation above.
To find the answers, we can use the properties of the normal distribution and z-scores. Assuming that the scores are normally distributed with a mean of 550 and a standard deviation of 100, we can calculate the z-scores for each condition and then use a z-table to find the corresponding percentages.
1. Greater than 600:
z = (x - μ) / σ
z = (600 - 550) / 100
z = 0.5
Using a z-table, we find that the area to the left of z = 0.5 is 0.6915. Since we need the area to the right of z = 0.5, we subtract this value from 1:
P(X > 600) = 1 - 0.6915 = 0.3085 (or 30.85%)
2. Greater than 750:
z = (x - μ) / σ
z = (750 - 550) / 100
z = 2
Using a z-table, we find that the area to the left of z = 2 is 0.9772. Since we need the area to the right of z = 2, we subtract this value from 1:
P(X > 750) = 1 - 0.9772 = 0.0228 (or 2.28%)
3. Less than 400:
z = (x - μ) / σ
z = (400 - 550) / 100
z = -1.5
Using a z-table, we find that the area to the left of z = -1.5 is 0.0668:
P(X < 400) = 0.0668 (or 6.68%)
4. Between 400 and 600:
For this condition, we need to find the area between two z-scores.
Finding the z-score for 400:
z1 = (400 - 550) / 100
z1 = -1.5
Finding the z-score for 600:
z2 = (600 - 550) / 100
z2 = 0.5
Using the z-table, we find the area to the left of z = -1.5 is 0.0668 and the area to the left of z = 0.5 is 0.6915. To find the area between these two z-scores, we subtract the smaller area from the larger one:
P(400 < X < 600) = 0.6915 - 0.0668 = 0.6247 (or 62.47%)
5. Q3 represents the third quartile and is the score at which 75% of the scores are below and 25% are above. Since the data is close to a normal distribution, we can calculate it using z-scores.
Using a z-table, we need to find the z-score that corresponds to the area of 0.75 (or 75th percentile). The closest z-score is 0.674.
Converting the z-score back to the original score:
x = z * σ + μ
x = 0.674 * 100 + 550
x ≈ 674 + 550
x ≈ 1224
Therefore, Q3 is approximately 1224.
6. The score that separates the top 70% from the bottom 30% represents the 70th percentile cutoff. Similar to the previous step, we can find the z-score that corresponds to the area of 0.70.
Using a z-table, the closest z-score is 0.524.
Converting the z-score back to the original score:
x = z * σ + μ
x = 0.524 * 100 + 550
x ≈ 52.4 + 550
x ≈ 602.4
Therefore, the score that separates the top 70% from the bottom 30% is approximately 602.4.
To find the answers to these questions, we can use the properties of the normal distribution and the given mean and standard deviation. Here's how:
1. To find the percentage of scores greater than 600:
First, we need to standardize the value 600 using the z-score formula: z = (x - mean) / standard deviation.
z = (600 - 550) / 100 = 0.5
Next, we look up the area to the right of the z-score 0.5 in the Standard Normal Distribution (z-table). The area is 0.3085 or 30.85%. So, approximately 30.85% of scores can be expected to be greater than 600.
2. To find the percentage of scores greater than 750:
Using the same steps as above:
z = (750 - 550) / 100 = 2
Looking up the area to the right of the z-score 2, we find it to be 0.0228 or 2.28%. Hence, approximately 2.28% of scores can be expected to be greater than 750.
3. To find the percentage of scores less than 400:
Again, using the z-score formula:
z = (400 - 550) / 100 = -1.5
Looking up the area to the left of the z-score -1.5, we find it to be 0.0668 or 6.68%. Therefore, approximately 6.68% of scores can be expected to be less than 400.
4. To find the percentage of scores between 400 and 600:
Since we already know the percentages of scores less than 400 and greater than 600, we can subtract those from 100% to find the percentage between. So, 100% - (6.68% + 30.85%) = 62.47%. Hence, approximately 62.47% of scores can be expected to be between 400 and 600.
Now, moving on to the additional questions:
5. What score represents Q3?
Q3 represents the third quartile or the 75th percentile. To find the corresponding score, we need to find the z-score for the 75th percentile using the z-table. The z-score that corresponds to the 75th percentile is approximately 0.675.
Using the z-score formula: z = (x - mean) / standard deviation, we can solve for x:
0.675 = (x - 550) / 100
Simplifying the equation, we get:
x - 550 = 67.5
x = 67.5 + 550
x = 617.5
So, the score representing 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 find the z-score that corresponds to the 70th percentile and then convert it back to the actual score.
Using the z-score formula: z = (x - mean) / standard deviation, we can solve for x:
z = 0.524 (approximate z-score for the 70th percentile from the z-table)
0.524 = (x - 550) / 100
Simplifying the equation, we get:
x - 550 = 52.4
x = 52.4 + 550
x = 602.4
Therefore, the score that separates the top 70% from the bottom 30% is approximately 602.4.
Please note that all calculations are approximate due to rounding.