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Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean ? = 520 and standard deviation ? = 115. a. Calculate the z-score for an SAT score of 720. Interpret it using a complete sentence. b. What math SAT score is 1.5 standard deviations above the mean? What can you say about this SAT score? c. For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. If one person took the SAT math test and scored 700 and a second person took the ACT math test and scored 30, who did better with respect to the test they took?

Step by step solution

01

Calculate the Z-score for an SAT Score of 720

The Z-score formula is \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the SAT score, \( \mu = 520 \) is the mean, and \( \sigma = 115 \) is the standard deviation. Substituting the values, we get \( z = \frac{720 - 520}{115} = \frac{200}{115} \approx 1.74 \).

02

Interpret the Z-score

A Z-score of approximately 1.74 means that an SAT score of 720 is 1.74 standard deviations above the mean. This indicates that a score of 720 is higher than the average SAT score.

03

Calculate the SAT Score 1.5 Standard Deviations above the Mean

To find the SAT score that is 1.5 standard deviations above the mean, use the formula \( x = \mu + (z \cdot \sigma) \). Here, \( \mu = 520 \), \( z = 1.5 \), and \( \sigma = 115 \). Substitute and calculate: \( x = 520 + (1.5 \cdot 115) = 520 + 172.5 = 692.5 \). Thus, the score is 692.5.

04

Analyze the SAT Score 1.5 Standard Deviations above the Mean

A score of 692.5, which is 1.5 standard deviations above the mean, indicates it is relatively higher than most of the population, as only about 6.68% of scores are expected to exceed 1.5 standard deviations in a normal distribution.

05

Calculate Z-scores for SAT and ACT Scores

For the SAT, use \( z = \frac{700 - 514}{117} = \frac{186}{117} \approx 1.59 \). For the ACT, use \( z = \frac{30 - 21}{5.3} = \frac{9}{5.3} \approx 1.70 \).

06

Interpret the Z-scores and Determine Who Did Better

The SAT Z-score of 1.59 is lower than the ACT Z-score of 1.70. Therefore, the person who took the ACT test scored relatively higher compared to the population, indicating they performed better with respect to their respective test.

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Key Concepts

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

Z-score

The Z-score is a measure that describes a data point's position relative to the mean of a group of values. It is expressed in terms of standard deviations from the mean. A Z-score allows us to see how unusual or typical a data point is within a normal distribution.

To calculate the Z-score, you use the formula: \[ z = \frac{x - \mu}{\sigma} \]where:- \( x \) is the individual score,- \( \mu \) is the mean of the distribution,- \( \sigma \) is the standard deviation.

For example, if you have an SAT score of 720, and the mean score is 520 with a standard deviation of 115, the Z-score would be: \[ z = \frac{720 - 520}{115} \approx 1.74 \]This Z-score indicates that the score of 720 is 1.74 standard deviations above the average SAT score. A positive Z-score means the score is above average, while a negative Z-score would indicate a score below average.

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It tells us how spread out the numbers are in a data set.

A smaller standard deviation means that values are closer to the mean, while a larger standard deviation indicates that the values are more spread out.

For instance, consider an SAT test with scores typically distributed around a mean. If the standard deviation is 115, this indicates that most scores are expected to fall within 115 points of the mean on either side. The formula for calculating standard deviation is:\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \]where:- \( N \) is the number of data points,- \( x_i \) represents each value in the data set,- \( \mu \) is the mean of the data set.

Standard deviation is a critical value when calculating Z-scores and understanding how much scores deviate from the average.

Mean Calculation

The mean, often referred to as the average, is a measure of central tendency that is calculated by adding all the individual values in a data set and then dividing by the number of values.

It provides a single value that summarizes the entire data set. The formula is:\[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i \]where:- \( N \) is the total number of data points,- \( x_i \) is each individual value in the data set.

For example, if you are calculating the average SAT math score when the scores are normally distributed, the mean might be 520. This tells us the central value at which scores are expected to cluster.

Understanding the mean is essential for calculating both the standard deviation and the Z-score, allowing for deeper statistical analysis.

Statistical Analysis

Statistical analysis is a process of collecting and analyzing data to identify patterns and trends. In the context of a normal distribution, statistical analysis can help to understand how different scores interact with the mean and standard deviation.

By comparing Z-scores, you can assess relative performance. For instance, comparing a Z-score of an SAT result with that of an ACT score enables you to see which is further from their respective means.

Statistical analysis also involves making predictions about data. Knowing that in a normal distribution, about 68% of data falls within one standard deviation of the mean, we can infer how rare or common certain values might be.

Understanding these statistical tools and concepts enables students to make informed conclusions about data sets in both academic settings and real-world applications. This involves using means, standard deviations, and Z-scores to interpret results effectively.