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Chapter 5: Problem 9

The mean score on the Stats exam was 75 points with a standard deviation of 5 points, and Gregor's z-score was - 2. How many points did he score?

Short Answer

Expert verified
Gregor scored 65 points.

Step by step solution

01

Understanding the Z-Score Formula

The z-score formula is given by \( z = \frac{X - \mu}{\sigma} \) where \( X \) is the score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
02

Plugging Values into the Formula

Given that Gregor's z-score \( z = -2 \), the mean \( \mu = 75 \), and the standard deviation \( \sigma = 5 \), we substitute these values into the formula: \(-2 = \frac{X - 75}{5}\).
03

Solving for Gregor's Exam Score (X)

To find \( X \), multiply both sides of the equation by 5: \(-2 \times 5 = X - 75\). Thus, \( -10 = X - 75\).
04

Finding the Final Score

Rearrange the equation \( -10 = X - 75 \) to solve for \( X \) by adding 75 to both sides: \( X = 65 \).

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Key Concepts

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

Understanding Standard Deviation
Standard deviation is a crucial concept in statistics. It measures the amount of variation or dispersion in a set of values.
For instance, when you have exam scores, the standard deviation can tell you how spread out the scores are around the mean score.
A low standard deviation means that most of the scores are close to the mean, while a high standard deviation indicates that the scores are spread out over a large range.
In the given problem, a standard deviation of 5 tells us that most students鈥� scores were within 5 points of the mean score of 75.
Calculating the Mean Score
The mean score, also known as the average, is another fundamental concept in statistics. It's calculated by adding up all the individual scores and then dividing by the total number of scores.
In the scenario where the mean was provided as 75, it tells us that when tallying up all exam scores together and dividing by the number of students, the average score was 75.
The mean is a useful measure because it gives an overall central value of your data, representing a middle point around which all data values cluster.
Using the Statistical Formula
The statistical formula used in the problem is the z-score formula, which is expressed as \( z = \frac{X - \mu}{\sigma} \).
The z-score itself indicates how many standard deviations an element is from the mean.
In simpler terms, a z-score tells us if a particular data point, like an exam score, is typical or unusual compared to the rest.
  • \( z \) is the z-score.
  • \( X \) is Gregor's actual exam score.
  • \( \mu \) (mu) is the mean score.
  • \( \sigma \) (sigma) is the standard deviation.
In the exercise, we can use this formula to find out Gregor鈥檚 actual exam score based on his z-score of -2.
Performing an Exam Score Calculation
Finding Gregor's exam score is an application of the z-score formula. Let's break it down.
Given Gregor's z-score as \(-2\), the mean as \(75\), and the standard deviation as \(5\), we start by inserting these values into the z-score formula:
\(-2 = \frac{X - 75}{5}\).
Next, we solve for \(X\), Gregor's actual score.
By multiplying both sides by 5, we balance the equation: \(-2 \times 5 = X - 75\) results in \(-10 = X - 75\).
Finally, add 75 to both sides to isolate \(X\): \(X = 65\).
Therefore, Gregor scored 65 on his exam, which is 10 points below the average of 75.

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Most popular questions from this chapter

One of the authors has an adopted grandson whose birth family members are very short. After examining him at his 2 -year checkup, the boy's pediatrician said that the \(z\) -score for his height relative to American 2-year-olds was \(-1.88 .\) Write a sentence explaining what that means.

Some IQ tests are standardized to a Normal model, with a mean of 100 and a standard deviation of 16 a) Draw the model for these IQ scores. Clearly label it, showing what the \(68-95-99.7\) Rule predicts. b) In what interval would you expect the central \(95 \%\) of IQ scores to be found? c) About what percent of people should have IQ scores above \(116 ?\) d) About what percent of people should have IQ scores between 68 and \(84 ?\) c) About what percent of people should have IQ scores above \(132 ?\)

People with \(z\) -scores above 2.5 on an IQ test are sometimes classified as geniuses. If IQ scores have a mean of 100 and a standard deviation of 16 points, what IQ score do you need to be considered a genius?

Most people think that the "normal" adult body temperature is \(98.6^{\circ} \mathrm{F}\). That figure, based on a 19th-century study, has recently been challenged. In a 1992 article in the Journal of the American Medical Association, researchers reported that a more accurate figure may be \(98.2^{\circ} \mathrm{F}\). Furthermore, the standard deviation appeared to be around \(0.7^{\circ} \mathrm{F}\). Assume that a Normal model is appropriate. a) In what interval would you expect most people's body temperatures to be? Explain. b) What fraction of people would be expected to have body temperatures above \(98.6^{\circ} \mathrm{F} ?\) c) Below what body temperature are the coolest \(20 \%\) of all people?

Hens usually begin laying eggs when they are about 6 months old. Young hens tend to lay smaller eggs, often weighing less than the desired minimum weight of 54 grams. a) The average weight of the eggs produced by the young hens is 50.9 grams, and only \(28 \%\) of their eggs exceed the desired minimum weight. If a Normal model is appropriate, what would the standard deviation of the egg weights be? b) By the time these hens have reached the age of 1 year, the eggs they produce average 67.1 grams, and \(98 \%\) of them are above the minimum weight. What is the standard deviation for the appropriate Normal model for these older hens? c) Are egg sizes more consistent for the younger hens or the older ones? Explain.
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