Scores on the GRE (Graduate Record Examination) are normally distributed with a mean of 541 and a standard deviation of 101. Use the 68-95-99.7 Rule to find the percentage of people taking the test who score between 238 and 844
The percentage of people taking the test who score betweenbetween 238 and 844 what %?
Z = (score-mean)/SD = (844-541)/101 = 303/101 = 3
Scores on the GRE (Graduate Record Examination) are normally distributed with a mean of 565 and a standard deviation of 82. Use the 68 dash 95 dash 99.7 Rule to find the percentage of people taking the test who score above 811.
The percentage of people taking the test who score above 811 is nothing %.
To find the percentage of people taking the test who score between 238 and 844, we can use the 68-95-99.7 Rule, also known as the empirical rule.
Step 1: Calculate the z-scores for the lower and upper limits.
The z-score formula is: z = (x - mean) / standard deviation
For the lower limit of 238:
z_lower = (238 - 541) / 101
For the upper limit of 844:
z_upper = (844 - 541) / 101
Step 2: Use a z-table or a calculator to find the percentage of the data between these z-scores.
The percentage between the lower and upper limits is the difference between the cumulative area to the left of the upper z-score and the cumulative area to the left of the lower z-score.
Let's use a z-table to find these cumulative areas.
The cumulative area to the left of the lower z-score can be found by looking up the z-score in the z-table. For z_lower = (238 - 541) / 101 = -3, the cumulative area is 0.0013.
The cumulative area to the left of the upper z-score can also be found by looking up the z-score in the z-table. For z_upper = (844 - 541) / 101 = 3, the cumulative area is 0.9987.
Step 3: Calculate the percentage between the lower and upper limits.
The percentage between the lower and upper limits is:
Percentage = (cumulative area to the left of the upper z-score) - (cumulative area to the left of the lower z-score)
Percentage = 0.9987 - 0.0013
Percentage = 0.9974
Therefore, the percentage of people taking the test who score between 238 and 844 is approximately 99.74%.
To find the percentage of people taking the test who score between 238 and 844, we can use the 68-95-99.7 Rule, also known as the empirical rule or the three-sigma rule. This rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, we are given that the mean is 541 and the standard deviation is 101. So, we need to find the percentage of scores that fall between 238 and 844, which is within two standard deviations of the mean.
To do this, we calculate the z-scores for both 238 and 844 using the formula:
z = (x - μ) / σ
Where:
- x is the given score (238 and 844 in this case),
- μ is the mean (541), and
- σ is the standard deviation (101).
For 238:
z = (238 - 541) / 101 ≈ -3.00
For 844:
z = (844 - 541) / 101 ≈ 3.00
Next, we use a z-table (or a calculator) to find the percentage of scores that fall between these two z-scores. Since the z-table usually provides the percentage for positive z-scores, we can subtract the percentage from 0.5 (representing the middle 50%) to get the percentage for both tails.
The percentage of scores that falls to the left of z = -3.00 is approximately 0.0013 (found on the z-table). Subtracting it from 0.5, we get 0.5 - 0.0013 = 0.4987 (approximately).
The percentage of scores that falls to the right of z = 3.00 is also approximately 0.0013. Subtracting it from 0.5, we get 0.5 - 0.0013 = 0.4987 (approximately).
Finally, we add both percentages together to get the total percentage of scores between 238 and 844:
0.4987 + 0.4987 = 0.9974
So, approximately 99.74% of people taking the test will score between 238 and 844.