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Chapter 3: Problem 40

Perfect SAT scores. It is possible to score higher than 1600 on the combined mathematics and reading portions of the SAT, but scores 1600 and above are reported as 1600 . The distribution of SAT scores (combining mathematics and reading) in 2014 was close to Normal with mean 1010 and standard deviation 218. What proportion of SAT scores for these two parts were reported as 1600 ? (That is, what proportion of SAT scores were actually higher than 1600?)

Short Answer

Expert verified
About 0.34% of SAT scores were reported as 1600.

Step by step solution

01

Understand the Problem

The problem asks us to find the proportion of SAT scores that are reported as 1600, meaning the scores that are actually higher than 1600. The SAT scores are normally distributed with a mean (\( \mu \)) of 1010 and a standard deviation (\( \sigma \)) of 218.
02

Convert the Score to a Z-Score

We need to find what's the probability of a score being more than 1600. Start by converting the score of 1600 to a z-score using the formula \( z = \frac{x - \mu}{\sigma} \). Here, \( x = 1600 \), \( \mu = 1010 \), \( \sigma = 218 \). Calculate:\[ z = \frac{1600 - 1010}{218} = \frac{590}{218} \approx 2.706 \].
03

Find the Proportion Above the Z-Score

Using the z-score obtained, \( z \approx 2.706 \), look up this value in a standard normal (z) distribution table or use a calculator to find the proportion that falls to the right of this z-score. This represents the proportion of scores higher than 1600.
04

Interpret the Z-Score Table

The z-score table gives us values to the left of the z-score. For \( z \approx 2.706 \), the value could be around 0.9966 (depending on the table). Since we want what's above 2.706, calculate \( 1 - 0.9966 = 0.0034 \).
05

Conclude the Proportion

The calculated value of 0.0034 means that about 0.34% of students scored above 1600 and therefore are reported as having a score of 1600.

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Key Concepts

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

Normal Distribution
The concept of a normal distribution is a fundamental idea in statistics. It describes how data points are distributed around mean values across many real-world scenarios. Imagine it as a symmetrical bell-shaped curve which indicates that most values cluster around a central region, known as the mean or average.
For normal distributions:
  • The mean (\( \mu \)) is located at the center of the curve, giving it that balanced shape.
  • The spread or variability of the data is denoted by the standard deviation (\( \sigma \)).
  • About 68% of data lies within one standard deviation from the mean, 95% within two, and 99.7% within three.
For the SAT scores mentioned, the mean is 1010 and the standard deviation is 218. Understanding normal distribution helps us visualize and calculate the probability of any score occurring within the range.
Z-Score Calculation
A z-score is a statistical tool that shows how many standard deviations an element is from the mean. It's crucial for comparing scores across different data sets or distributions.
To calculate the z-score, you use the formula:\[z = \frac{x - \mu}{\sigma}\]Where:
  • \( x \) is the score you want to standardize.
  • \( \mu \) is the mean of the distribution.
  • \( \sigma \) is the standard deviation.
For instance, if a student scored 1600 on the SAT, the z-score would be calculated as:\[z = \frac{1600 - 1010}{218} \approx 2.706\]This value indicates that a score of 1600 is 2.706 standard deviations above the mean. The higher this value, the rarer the score.
Proportion Calculation
Once you have the z-score, you can determine the proportion of data points that lie either above or below this score in a normal distribution. This involves checking a standard normal distribution (z) table.
The z-table gives proportions to the left of a given z-score. For our calculation with a z-score of approximately 2.706, the table might show a value like 0.9966.
To find the proportion of scores above this z-score (or higher than 1600 in our example), subtract this value from 1:\[1 - 0.9966 = 0.0034\]This result, 0.0034, means about 0.34% of students scored above 1600. Knowing these proportions is important for interpreting how unusual or likely certain outcomes are within a data set.
Statistics Education
Understanding basic statistical concepts like normal distribution, z-score calculations, and proportion calculations is essential in today's data-driven world. These skills not only aid in interpreting data but also in making informed decisions based on statistical evidence.
In statistics education, students learn to:
  • Identify and describe different types of data distribution.
  • Calculate and interpret z-scores for standardization of data.
  • Utilize statistical tables and tools for proportion calculations.
These statistical tools are widely used in fields from psychology and education to business and healthcare. A solid foundation in statistics can empower students to critically analyze data trends and probabilistic outcomes effectively.

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Most popular questions from this chapter

Monsoon Rains. The summer monsoon rains bring \(80 \%\) of India's rainfall and are essential for the country's agriculture. Records going back more than a century show that the amount of monsoon rainfall varies from year to year according to a distribution that is approximately Normal with mean 852 millimeters \((\mathrm{mm})\) and standard deviation \(82 \mathrm{~mm}^{3}\) Use the \(68-95-99.7\) rule to answer the following questions. (a) Between what values do the monsoon rains fall in the middle \(95 \%\) of all years? (b) How small are the monsoon rains in the driest \(2.5 \%\) of all years?

The Medical College Admissions Test. A new version of the Medical College Admissions Test (MCAT) was introduced in spring 2015 and is intended to shift the focus from what applicants know to how well they can use what they know. One result of the change is that the scale on which the exam is graded has been modified, with the total score of the four sections on the test ranging from 472 to 528 . In spring 2015 , the mean score was \(500.0\) with a standard deviation of \(10.6\) (a) What are the median and the first and third quartiles of the MCAT scores? What is the interquartile range? (b) Give the interval that contains the central \(80 \%\) of the MCAT scores.

Use the Normal Table. Use Table A to find the proportion of observations from a standard Normal distribution that satisfies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the question. (a) \(z<-0.42\) (b) \(z>-1.58\) (c) \(z<2.12\) (d) \(-0.42

Acid rain? Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by \(\mathrm{pH}\) on a scale of 0 to 14 . Distilled water has \(\mathrm{pH} 7.0\), and lower \(\mathrm{pH}\) values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below \(5.0\). The \(\mathrm{pH}\) of rain at one location varies among rainy days according to a Normal distribution with mean \(5.43\) and standard deviation \(0.54\). What proportion of rainy days have rainfall with \(\mathrm{pH}\) below \(5.0\) ?

Fruit flies. The common fruit fly Drosophila melanogaster is the most studied organism in genetic research because it is small, is easy to grow, and reproduces rapidly. The length of the thorax (where the wings and legs attach) in a population of male fruit flies is approximately Normal with mean \(0.800\) millimeter (mm) and standard deviation \(0.078 \mathrm{~mm}\). (a) What proportion of flies have thorax length less than \(0.7 \mathrm{~mm}\) ? (b) What proportion have thorax length greater than \(1.0 \mathrm{~mm}\) ? (c) What proportion have thorax length between \(0.7 \mathrm{~mm}\) and \(1.0 \mathrm{~mm}\) ?
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