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Chapter 6: Problem 4

The national average SAT score (for Verbal and Math) is 1028 . If we assume a normal distribution with \(\sigma=92,\) what is the 90th percentile score? What is the probability that a randomly selected score exceeds \(1200 ?\)

Short Answer

Expert verified
The 90th percentile score is approximately 1146, and the probability of exceeding 1200 is about 3.07%.

Step by step solution

01

Understanding the Problem

We have a normal distribution of SAT scores with a mean (5) of 1028 and standard deviation (85) of 92. We need to find the 90th percentile score, which is the score below which 90% of the scores fall. We also need to calculate the probability of a randomly selected score exceeding 1200.
02

Finding the 90th Percentile

The 90th percentile is the score 'x' such that 90% of the scores are below 'x'. For a standard normal distribution, a z-score for the 90th percentile is approximately 1.28. Using the formula for converting between a score and a z-score: \[ x = 5 + z imes 85 \]where \( z = 1.28, \mu = 1028, \sigma = 92 \). Calculate:\[ x = 1028 + 1.28 imes 92 \]\[ x 1145.76 \]Therfore, the 90th percentile score is approximately 1146.
03

Calculating Probability of a Score Exceeding 1200

To find the probability that a score exceeds 1200, we first calculate the z-score for 1200:\[ z = \frac{(1200 - 1028)}{92} \]\[ z 1.87 \]Using the standard normal distribution table or calculator, the cumulative probability to the left of z=1.87 is approximately 0.9693. Since we want the probability of scoring more than 1200, we subtract this value from 1:\[ P(X > 1200) = 1 - 0.9693 = 0.0307 \]Thus, there's a 3.07% probability of a randomly selected score exceeding 1200.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

SAT Scores
SAT scores are a crucial aspect of the college application process. They are designed to assess a student's readiness for college-level work. The scores from the SAT are used by schools to help decide a student's academic potential. The SAT covers two main areas: Verbal and Math.
  • Verbal: Assesses reading and writing skills.
  • Math: Measures mathematical skills acquired up to the beginning of grade 12.
The national average SAT score provides a benchmark against which individual performances can be compared. In our exercise, the average is 1028, indicating where most students typically score. Understanding the distribution of these scores can help in understanding how an individual score compares to the average.
Percentiles
A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For SAT scores, knowing which percentile corresponds to a particular score can provide insight into how a score ranks among all test-takers.
  • The 90th percentile means that 90% of the scores fall below this value.
  • In our scenario, calculating the 90th percentile determines what score a student needs to outperform 90% of their peers.
To find the 90th percentile score from a normal distribution, we use the concept of z-scores, another statistical measure that converts raw scores into a standard form.
Z-score
The z-score is a statistical measurement that describes a value's relation to the mean of a group of values. If a z-score is 0, it indicates that the data point's score is identical to the mean score. The z-score can help determine how far away a score is from the average and how unusual it is.
  • The formula: \[ z = \frac{(X - \mu)}{\sigma} \]
  • Where \( X \) is the score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
In our example, the z-score tells us how a score like 1200 compares to other scores in the distribution. Doing this calculation for both the 90th percentile and for scores greater than 1200 gives context to individual performance. Calculating the z-score helps us find probabilities associated with certain scores.
Probability Calculation
Probability calculations help us understand the likelihood of a score occurring within a certain range. This is done using the area under the normal curve, which represents probabilities.
  • To find the probability of a randomly selected score exceeding 1200, we calculate its z-score.
  • This z-score then indicates the position of the score within the distribution.
  • Using a z-table, or a similar tool, calculates the cumulative probability up to this z-score.
  • To find the probability of exceeding this score, subtract this cumulative probability from 1.
In our example, a score of 1200 corresponds to a z-score of 1.87. From the standard normal distribution table, the cumulative probability for a z-score of 1.87 is approximately 0.9693. Thus, the chance of scoring higher than 1200 is about 3.07%, offering useful insights into where a particular score stands in comparison to the national average.

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