Assume that GRE scores approximate a normal curve and have mean of 500 points and a standard deviation of 100. What proportion of the normal curve would correspond to each of the following situations?
1) A score more than 50 points above the mean?
2) A score more than 100 points either above or below the mean?
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To find the proportion of the normal curve that corresponds to each of the given situations, we need to calculate the area under the normal curve.
1) A score more than 50 points above the mean:
To calculate the proportion of the normal curve corresponding to a score more than 50 points above the mean, we need to find the area to the right of the given value in the standard normal distribution.
First, we need to convert the given value into a z-score, which represents the number of standard deviations the value is from the mean. We can calculate the z-score using the formula:
z = (x - μ) / σ,
where z is the z-score, x is the given value, μ is the mean, and σ is the standard deviation.
In this case, the given value is 50 points above the mean (μ = 500), so x = 500 + 50 = 550. The standard deviation is σ = 100.
Calculating the z-score:
z = (550 - 500) / 100 = 50 / 100 = 0.5.
Now, we can find the proportion of the normal curve to the right of the z-score using a standard normal distribution table or a statistical calculator. For a z-score of 0.5, the area to the right is approximately 0.3085.
Therefore, the proportion of the normal curve corresponding to a score more than 50 points above the mean is approximately 0.3085 or 30.85%.
2) A score more than 100 points either above or below the mean:
To find the proportion of the normal curve corresponding to a score more than 100 points either above or below the mean, we need to consider both the areas to the right of 100 points above the mean and to the left of 100 points below the mean.
For the area to the right of 100 points above the mean, we can follow the same steps as in the first situation. Calculating the z-score:
z = (500 + 100 - 500) / 100 = 100 / 100 = 1.
Using the standard normal distribution table or calculator, the area to the right of a z-score of 1 is approximately 0.1587.
For the area to the left of 100 points below the mean, we calculate the z-score:
z = (500 - 100 - 500) / 100 = -100 / 100 = -1.
Using the standard normal distribution table or calculator, the area to the left of a z-score of -1 is also approximately 0.1587.
Now, we can add these two areas together to get the proportion of the normal curve corresponding to a score more than 100 points either above or below the mean:
0.1587 + 0.1587 = 0.3174.
Therefore, the proportion of the normal curve corresponding to a score more than 100 points either above or below the mean is approximately 0.3174 or 31.74%.