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Chapter 3: Problem 10

Consider a value to be significantly low if its \(z\) score is less than or equal to -2 or consider the value to be significantly high if its \(z\) score is greater than or equal to \(2 .\) In a recent year, scores on the Medical College Admission Test (MCAT) had a mean of 25.2 and a standard deviation of \(6.4 .\) Identify the MCAT scores that are significantly low or significantly high.

Short Answer

Expert verified
MCAT scores \(\leq 12.4\) are significantly low and scores \(\geq 38\) are significantly high.

Step by step solution

01

Recall the Formula for Z-Score

The z-score formula is used to determine how many standard deviations away a value is from the mean. The formula is given by: \[ z = \frac{x - \text{mean}}{\text{standard deviation}} \] Where: \(x\) is the value of the score, the \(\text{mean}\) is the average score, and the \(\text{standard deviation}\) measures the typical distance of scores from the mean.
02

Define Criteria for Significantly Low and High Scores

A score is considered significantly low if its z-score is less than or equal to \(-2\). A score is considered significantly high if its z-score is greater than or equal to \(2\).
03

Calculate the Threshold for Significantly Low Scores

We'll start by finding the score that corresponds to a z-score of \(-2\). Using the formula: \[ x = (z \times \text{standard deviation}) + \text{mean} \] For \(z = -2\), substituting the values we have: \[ x = (-2 \times 6.4) + 25.2 = -12.8 + 25.2 = 12.4 \] Therefore, any score \(\leq 12.4\) is considered significantly low.
04

Calculate the Threshold for Significantly High Scores

Next, we'll find the score that corresponds to a z-score of \(2\). Again using the formula: \[ x = (z \times \text{standard deviation}) + \text{mean} \] For \(z = 2\), substituting the values we have: \[ x = (2 \times 6.4) + 25.2 = 12.8 + 25.2 = 38 \] Therefore, any score \(\geq 38\) is considered significantly high.
05

Summarize the Findings

Scores on the MCAT are considered significantly low if they are \(\leq 12.4\) and significantly high if they are \(\geq 38\).

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Key Concepts

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

z-score
A z-score helps us understand how far a specific score is from the average value. It tells us how many standard deviations a value is away from the mean. To calculate it, we use the formula: \[ z = \frac{x - \text{mean}}{\text{standard deviation}} \]
  • x is the value of the score.
  • The mean is the average score.
  • The standard deviation shows the typical amount by which scores differ from the mean.
So, if a score has a high z-score, it's far from the mean and unusual. If it's low, it's close to the mean and typical. Breaking down this concept brings clarity and helps you see the bigger picture.
significantly low scores
Significantly low scores are those that stand out as extremely below average. In statistics, a score is considered significantly low if its z-score is \[ \text{z} \text{\鈮 -2 \]This means the score is two or more standard deviations below the mean. To find such scores, use the formula:\[ x = (z \times \text{standard deviation}) + \text{mean} \]For example, with a mean score of 25.2 and a standard deviation of 6.4 in MCAT, any score \text\鈮� 12.4\ would be significantly low. These scores are uncommon and point to underperformance.
significantly high scores
Significantly high scores are those that are much higher than the average. We consider these scores special because they stand out. A score is significantly high if its z-score is:\[ \text{z} \text{ \鈮 2 \]This means the score is at least two standard deviations above the mean. Using the formula: \[ x = (z \times \text{standard deviation}) + \text{mean} \]For the MCAT, with a mean score of 25.2 and standard deviation of 6.4, any score \鈮� 38\ is significantly high. These scores imply a strong performance and are relatively rare.
standard deviation
Standard deviation is a key concept in statistics that measures the spread of scores around the mean. It tells us how much scores typically differ from the average. A smaller standard deviation indicates that the scores tend to be close to the mean, while a larger one indicates greater variability.
  • To calculate significantly low or high scores, standard deviation is crucial.
  • It helps us determine how unusual a score is in a dataset.
For instance, with a standard deviation of 6.4 in the MCAT scores, we know that most scores fall within 6.4 points of the mean of 25.2.
mean
The mean, or average, is the central value of a dataset. It's calculated by adding up all the scores and dividing by the number of scores. The mean provides a reference point for measuring how individual scores vary.
  • In the context of z-scores, the mean helps us understand if a score is higher or lower than average.
  • For the MCAT, the mean score is 25.2.
Knowing the mean allows us to calculate z-scores, which then help in identifying significantly low or high scores. It is a fundamental concept for any statistical analysis.

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Most popular questions from this chapter

Find the coefficient of variation for each of the two samples; then compare the variation. (The same data were used in Section 3-I.) Listed below are amounts (in millions of dollars) collected from parking meters by Brinks and others in New York City during similar time periods. A larger data set was used to convict five Brinks employees of grand larceny. The data were provided by the attorney for New York City, and they are listed on the DASL Website. Do the two samples appear to have different amounts of variation? $$\begin{array}{lcccccccccc} \text { Collection Contractor Was Brinks } & 1.3 & 1.5 & 1.3 & 1.5 & 1.4 & 1.7 & 1.8 & 1.7 & 1.7 & 1.6 \\ \text { Collection Contractor Was Not Brinks } & 2.2 & 1.9 & 1.5 & 1.6 & 1.5 & 1.7 & 1.9 & 1.6 & 1.6 & 1.8 \end{array}$$

Identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance.

Find the mean and median for each of the two samples, then compare the two sets of results. Waiting times (in seconds) of customers at the Madison Savings Bank are recorded with two configurations: single customer line; individual customer lines. Carefully examine the data to determine whether there is a difference between the two data sets that is not apparent from a comparison of the measures of center. If so, what is it? $$\begin{array}{llllllllll}\text { single Line } & 390 & 396 & 402 & 408 & 426 & 438 & 444 & 462 & 462 & 462 \\\\\text { Individual Lines } & 252 & 324 & 348 & 372 & 402 & 462 & 462 & 510 & 558 & 600\end{array}$$

Use the given data to construct a boxplot and identify the 5-number summary. Blood Pressure Measurements Fourteen different second-year medical students at Bellevue Hospital measured the blood pressure of the same person. The systolic readings (mm Hg) are listed below. $$\begin{array}{rrrrrrrrrrrrrr} 138 & 130 & 135 & 140 & 120 & 125 & 120 & 130 & 130 & 144 & 143 & 140 & 130 & 150 \end{array}$$

The quadratic mean (or root mean square, or R.M.S.) is used in physical applications, such as power distribution systems. The quadratic mean of a set of values is obtained by squaring each value, adding those squares, dividing the sum by the number of values \(n,\) and then taking the square root of that result, as indicated below: $$\text { Quadratic mean }=\sqrt{\frac{\Sigma x^{2}}{n}}$$ Find the R.M.S. of these voltages measured from household current: 0,60,110,-110,-60,0 How does the result compare to the mean?
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