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Scholastic Assessment Test (SAT) scores, which have fallen slowly since the inception of the test, have now begun to rise. Originally, a score of 500 was intended to be average. The mean scores for 2005 were approximately 508 for the verbal test and 520 for the mathematics test. A random sample of the test scores of 20 seniors from a large urban high school produced the means and standard deviations listed in the accompanying table: $$\begin{array}{lcc} \hline & \text { Verbal } & \text { Mathematics } \\ \hline \text { Sample mean } & 505 & 495 \\ \text { Sample standard deviation } & 57 & 69 \\ \hline \end{array}$$ a. Find a \(90 \%\) confidence interval for the mean verbal SAT scores for high school seniors from the urban high school. b. Does the interval that you found in part (a) include the value 508 , the true mean verbal SAT score for \(2005 ?\) What can you conclude? c. Construct a \(90 \%\) confidence interval for the mean mathematics SAT score for the urban high school seniors. Does the interval include 520 , the true mean mathematics score for \(2005 ?\) What can you conclude?

Step by step solution

01

Understanding Confidence Interval Concept

To find a confidence interval for the mean, we use the formula \( \bar{x} \pm z \frac{s}{\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( z \) is the z-score, \( s \) is the sample standard deviation, and \( n \) is the sample size. Here, \( z \) is the critical value for a 90% confidence level.

02

Determine Critical Value

For a 90% confidence interval, the critical value \( z \) is approximately 1.645 from the standard normal distribution table.

03

Calculate Verbal SAT Confidence Interval

For verbal scores, \( \bar{x} = 505 \), \( s = 57 \), and \( n = 20 \). Substitute these into the formula: \( 505 \pm 1.645 \times \frac{57}{\sqrt{20}} \). Compute \[ \frac{57}{\sqrt{20}} \approx 12.75 \]. Then, \( 1.645 \times 12.75 \approx 20.98 \). The confidence interval is \([505 - 20.98, 505 + 20.98]\), or \([484.02, 525.98]\).

04

Analyze Verbal Interval and Conclusion

Check if 508 falls within the computed interval \([484.02, 525.98]\). Since 508 is within this interval, we cannot conclude that the mean verbal SAT score for this high school is different from the 2005 average.

05

Calculate Mathematics SAT Confidence Interval

For mathematics scores, \( \bar{x} = 495 \), \( s = 69 \), and \( n = 20 \). Substitute into the formula: \( 495 \pm 1.645 \times \frac{69}{\sqrt{20}} \). Compute \[ \frac{69}{\sqrt{20}} \approx 15.43 \]. Then, \( 1.645 \times 15.43 \approx 25.37 \). The confidence interval is \([495 - 25.37, 495 + 25.37]\), or \([469.63, 520.37]\).

06

Analyze Mathematics Interval and Conclusion

Check if 520 falls within the computed interval \([469.63, 520.37]\). Since 520 is within this interval, we cannot conclude that the mean mathematics SAT score for this high school is different from the 2005 average.

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Key Concepts

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

SAT Scores

SAT scores, known as Scholastic Assessment Test scores, are standardized test results widely used for college admissions in the United States. They aim to measure a student's readiness for college by assessing skills in verbal and mathematical areas. Historically, a score of 500 on each section was established as the average score, allowing colleges to compare applicants from different educational backgrounds. In 2005, the mean verbal score jumped to approximately 508, while the mean mathematics score was around 520. These averages give insight into the national performance trends of students during that time. SAT scores are crucial for educational institutions because they provide a common data point used to evaluate the aptitude and potential of students, particularly when comparing individuals from diverse high schools and curriculums.

Mean

The concept of the mean, or average, is a fundamental statistical measure that is widely used to describe a set of data. It provides a central value around which the data points are distributed. To calculate the mean, sum up all the scores in a dataset and then divide by the number of data points. For instance, if we take the sample scores for the verbal and mathematics tests from our exercise, we found sample means of 505 and 495, respectively, for these sections from a group of 20 students.
The mean helps in understanding the overall performance of a sample and establishing benchmarks or standards. It acts as an indicator of what would be considered typical within the group being studied. In the context of SAT scores, mean values are used to compare the academic aptitude of students against national averages, aiding in assessing educational progress over time.

Standard Deviation

Standard deviation is a measure of how dispersed the data points are relative to the mean. It provides an indication of the variability or spread of a data set. The standard deviation can tell us how much individual SAT scores tend to deviate from the average SAT score.
In statistics, a lower standard deviation suggests that data points are closer to the mean, indicating less variability. Conversely, a higher standard deviation indicates more spread and variability in the scores. In the exercise, the standard deviation for the verbal SAT scores was 57, while for the mathematics SAT scores, it was 69, indicating some level of variability in the students’ performance on these tests.
Standard deviation is crucial in constructing confidence intervals because it helps determine the range around the mean where the true mean of the population is likely to fall.

Sample Size

Sample size refers to the number of observations or data points in a sample. It plays a critical role in the accuracy and reliability of statistical analysis. A larger sample size usually offers a more accurate representation of the population, leading to more reliable results, while a smaller sample size might lead to less reliable conclusions, mainly due to increased variability.
In the context of the SAT score exercise, the sample size was 20, meaning the data was drawn from 20 students. This size affects the confidence interval width; a smaller sample size, as in this case, might lead to a wider confidence interval. The logic is simple: a larger sample size would reduce errors and provide more precise estimates of the population mean, assuming that the sample is representative.