Scores on the GRE ( graduate record examination) are normally distributed with a mean of 536 and a standard deviation of 76. Use the 68-95-99.7. Rule to find the percentage of people taking the test who score between 460 and 536
The percentage of people taking the test whos scores between 460 and 536 is what percentage?
Z = (score-mean)/SD = (460-536)/76 = -76/76 = -1
Looking it up on a table in the back of your text, between mean and one SD below the mean = 34%
460 is one s.d. below the mean of 536
this makes it half of the "68" ... so, 34%
To find the percentage of people taking the test who score between 460 and 536, we can use the z-score formula:
z = (x - μ) / σ
Where:
x = the specific score (460 or 536 in this case)
μ = the mean of the distribution (536)
σ = the standard deviation (76)
For the lower bound (460), the z-score is:
z = (460 - 536) / 76 = -1
For the upper bound (536), the z-score is:
z = (536 - 536) / 76 = 0
Now, using the 68-95-99.7 Rule:
- Approximately 68% of the scores fall within 1 standard deviation of the mean.
- Approximately 95% of the scores fall within 2 standard deviations of the mean.
- Approximately 99.7% of the scores fall within 3 standard deviations of the mean.
Since the z-scores for the lower bound (z = -1) and upper bound (z = 0) fall within 1 standard deviation of the mean, the percentage of people who score between 460 and 536 is approximately 68%.
To find the percentage of people taking the test who score between 460 and 536, we can use the 68-95-99.7 Rule, also known as the Empirical Rule or the Three Sigma Rule.
According to this rule, in 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, the mean is 536 and the standard deviation is 76.
To find the percentage of people who score between 460 and 536, we need to calculate the z-scores for these two values.
The formula to calculate the z-score is:
z = (x - μ) / σ
where:
- x is the value we want to find the z-score for (in this case, 460 and 536),
- μ is the mean of the distribution (536), and
- σ is the standard deviation of the distribution (76).
For 460:
z1 = (460 - 536) / 76 = -1.00
For 536:
z2 = (536 - 536) / 76 = 0.00
Now, we can use the standard normal distribution table or a calculator to find the corresponding area under the curve for these z-scores.
The area between these two z-scores represents the percentage of people who score between 460 and 536.
By referring to the standard normal distribution table or using a calculator, we find that the area to the left of z1 (-1.00) is approximately 0.1587, and the area to the left of z2 (0.00) is approximately 0.5000.
To find the area between z1 and z2, we subtract the smaller area from the larger area:
0.5000 - 0.1587 = 0.3413
Finally, to find the percentage, we multiply this decimal value by 100:
0.3413 * 100 = 34.13%
Therefore, approximately 34.13% of the people taking the test will score between 460 and 536.