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The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students’ scores on the quantitative portion of the GRE follow a normal distribution with a mean of 150 and a standard deviation of 8.8.

1. What proportion of GRE scores can be expected to be over 160?

2. What proportion of GRE scores can be expected to be under 160?

3. What proportion of GRE scores can be expected to be between 155 and 160?

4. What is the probability that a randomly selected student will score over 145 points?

5. What is the probability that a randomly selected student will score less than 150 points?

6. What is the percentile rank of a student who earns a quantitative GRE score of 142?

Answer :

The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students’ scores on the quantitative portion of the GRE follow a normal distribution with mean 150 and standard deviation 8.8.A graduate school requires that students score above 160 to be admitted.

Proportion of combined GRE scores can be expected to be over 160:We are given that the mean is 150 and the standard deviation is 8.8. We have to calculate the proportion of combined GRE scores that can be expected to be over 160.The standardized score is calculated as:z = (x - μ) / σwhere x = 160, μ = 150, and σ = 8.8Then we have:z = (160 - 150) / 8.8z = 1.136The area under the standard normal distribution curve to the right of 1.136 is 0.127. This means that 12.7% of combined GRE scores can be expected to be over 160.Proportion of combined GRE scores can be expected to be under 160:To calculate the proportion of combined GRE scores that can be expected to be under 160, we can subtract the proportion that is over 160 from the total proportion, which is 1.

So, the proportion of combined GRE scores that can be expected to be under 160 is:1 - 0.127 = 0.873This means that 87.3% of combined GRE scores can be expected to be under 160.Proportion of combined GRE scores can be expected to be between 155 and 160:We can use the same formula to calculate the proportion of combined GRE scores that can be expected to be between 155 and 160. First, we need to calculate the standardized scores for 155 and 160.z1 = (155 - 150) / 8.8z1 = 0.568z2 = (160 - 150) / 8.8z2 = 1.136Then, we need to find the area under the standard normal distribution curve between these two standardized scores.Using a standard normal distribution table or calculator, we find that the area between z = 0.568 and z = 1.136 is 0.155.

Therefore, the proportion of combined GRE scores that can be expected to be between 155 and 160 is 0.155. This means that 15.5% of combined GRE scores can be expected to be between 155 and 160.What is the probability that a randomly selected student will score over 145 points?We are given that the mean is 150 and the standard deviation is 8.8. We have to calculate the probability that a randomly selected student will score over 145 points.The standardized score is calculated as:z = (x - μ) / σwhere x = 145, μ = 150, and σ = 8.8Then we have:z = (145 - 150) / 8.8z = -0.568The area under the standard normal distribution curve to the right of -0.568 is 0.715. This means that the probability that a randomly selected student will score over 145 points is 0.715.

In summary, we can expect that 12.7% of combined GRE scores will be over 160, and 87.3% of combined GRE scores will be under 160. The proportion of combined GRE scores that can be expected to be between 155 and 160 is 15.5%. A randomly selected student has a probability of 0.715 of scoring over 145 points and a probability of 0.5 of scoring less than 150 points. Finally, a student who earns a quantitative GRE score of 142 has a percentile rank of 18.2%. These calculations are based on the normal distribution of GRE scores with a mean of 150 and a standard deviation of 8.8.

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