I do not understand this homework at all. Any help or elaboration would be greatly appreciated. Thanks.
100 juniors at Southwest High took the SAT test. The scores were distributed normally with a mean of 22 and a standard deviation of 3. Label the mean and three standard deviations from the mean.
a) What percentage of scores are between 22 and 25?
b) What percentage of scores are between scores 16 and 28?
c) What percentage of scores is less than a score of 13?
d) What percentage of scores is greater than 25?
e) Approximately how many juniors scored higher than 22?
f) Approximately how many juniors scored between 19 and 25?
you can get an idea of how these things work here:
http://davidmlane.com/hyperstat/z_table.html
try using your numbers, and you will see shaded areas.
To solve this problem, we need to use the Z-score formula. The Z-score measures how many standard deviations an observation or data point is from the mean. It can be calculated using the following formula:
Z = (X - μ) / σ
Where:
- X is the data point or observation
- μ is the mean
- σ is the standard deviation
Using this formula, we can find the Z-scores for different data points and then use the Z-table or a statistical calculator to find the corresponding probabilities.
a) To find the percentage of scores between 22 and 25, we need to find the Z-score for both values and then look up the corresponding probabilities in the Z-table.
Z1 = (22 - 22) / 3 = 0
Z2 = (25 - 22) / 3 = 1
Looking up the Z-scores in the Z-table, we find that the probability for a Z-score of 0 is 0.5 and the probability for a Z-score of 1 is 0.8413. To find the percentage of scores between 22 and 25, we subtract the probability for a Z-score of 0 from the probability for a Z-score of 1 and multiply by 100.
Percentage = (0.8413 - 0.5) * 100 = 34.13%
Therefore, approximately 34.13% of the scores are between 22 and 25.
b) To find the percentage of scores between 16 and 28, we need to find the Z-scores for both values and then look up the corresponding probabilities in the Z-table.
Z1 = (16 - 22) / 3 = -2
Z2 = (28 - 22) / 3 = 2
Looking up the Z-scores in the Z-table, we find that the probability for a Z-score of -2 is 0.0228 and the probability for a Z-score of 2 is 0.9772. To find the percentage of scores between 16 and 28, we subtract the probability for a Z-score of -2 from the probability for a Z-score of 2 and multiply by 100.
Percentage = (0.9772 - 0.0228) * 100 = 95.44%
Therefore, approximately 95.44% of the scores are between 16 and 28.
c) To find the percentage of scores less than a score of 13, we need to find the Z-score for 13 and then look up the corresponding probability in the Z-table.
Z = (13 - 22) / 3 = -3
Looking up the Z-score in the Z-table, we find that the probability for a Z-score of -3 is 0.0013.
Therefore, approximately 0.13% of the scores are less than 13.
d) To find the percentage of scores greater than 25, we need to find the Z-score for 25 and then subtract the probability from 1.
Z = (25 - 22) / 3 = 1
Looking up the Z-score in the Z-table, we find that the probability for a Z-score of 1 is 0.8413.
Percentage = (1 - 0.8413) * 100 = 15.87%
Therefore, approximately 15.87% of the scores are greater than 25.
e) To find the approximate number of juniors who scored higher than 22, we need to find the Z-score for 22 and subtract the result from 1 (representing the entire distribution).
Z = (22 - 22) / 3 = 0
Since the Z-score is 0, we can directly say that 50% of the scores are higher than 22.
Therefore, approximately 50% of the juniors scored higher than 22.
f) To find the approximate number of juniors who scored between 19 and 25, we need to find the Z-scores for both values and subtract the Z-score for 19 from the Z-score for 22.
Z1 = (19 - 22) / 3 = -1
Z2 = (25 - 22) / 3 = 1
Using the Z-table, we find that the probability for a Z-score of -1 is 0.1587, and the probability for a Z-score of 1 is 0.8413. To find the percentage of scores between 19 and 25, we subtract the probability for a Z-score of -1 from the probability for a Z-score of 1.
Percentage = (0.8413 - 0.1587) * 100 = 68.26%
Therefore, approximately 68.26% of the juniors scored between 19 and 25.