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Chapter 6: Problem 9

The college physical education department offered an advanced first aid course last semester. The scores on the comprehensive final exam were normally distributed, and the \(z\) scores for some of the students are shown below: Robert, 1.10 Joel, 0.00 Juan, 1.70 Jan, -0.80 Susan, -2.00 Linda, 1.60 (a) Which of these students scored above the mean? (b) Which of these students scored on the mean? (c) Which of these students scored below the mean? (d) If the mean score was \(\mu=150\) with standard deviation \(\sigma=20,\) what was the final exam score for each student?

Short Answer

Expert verified
(a) Robert, Juan, Linda scored above; (b) Joel scored on; (c) Jan, Susan scored below; Scores: Robert 172, Joel 150, Juan 184, Jan 134, Susan 110, Linda 182.

Step by step solution

01

Understanding Z-scores

Z-scores represent the number of standard deviations a data point is from the mean. A positive z-score indicates a score above the mean, a z-score of zero indicates a score exactly at the mean, and a negative z-score indicates a score below the mean.
02

Identify Scores Above the Mean

Students scoring above the mean have positive z-scores. Robert, with a z-score of 1.10, Juan, with 1.70, and Linda, with 1.60, scored above the mean.
03

Identify Scores On the Mean

A z-score of 0.00 represents a score exactly at the mean. Joel has a z-score of 0.00, indicating he scored on the mean.
04

Identify Scores Below the Mean

Students scoring below the mean have negative z-scores. Jan, with a z-score of -0.80, and Susan, with -2.00, scored below the mean.
05

Calculate Robert's Score

Use the z-score formula for Robert, where his z-score is 1.10, mean  is 150, and standard deviation  is 20: \[x =  + z imes  = 150 + 1.10 imes 20 = 172\] Robert's score is 172.
06

Calculate Joel's Score

For Joel, his z-score is 0.00: \[x =  + z imes  = 150 + 0.00 imes 20 = 150\] Joel's score is 150.
07

Calculate Juan's Score

For Juan, with a z-score of 1.70: \[x =  + z imes  = 150 + 1.70 imes 20 = 184\] Juan's score is 184.
08

Calculate Jan's Score

For Jan, with a z-score of -0.80: \[x =  + z imes  = 150 + (-0.80) imes 20 = 134\] Jan's score is 134.
09

Calculate Susan's Score

For Susan, with a z-score of -2.00: \[x =  + z imes  = 150 + (-2.00) imes 20 = 110\] Susan's score is 110.
10

Calculate Linda's Score

For Linda, with a z-score of 1.60: \[x =  + z imes  = 150 + 1.60 imes 20 = 182\] Linda's score is 182.

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Key Concepts

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

Normal Distribution
A normal distribution, often referred to as a bell curve due to its shape, is a statistical distribution where most of the data points cluster around the mean.
This pattern means that the majority of scores fall near the average value, with fewer scores appearing as you move further away from it.
  • Symmetrical: The left and right sides of the curve are mirror images.
  • Mean, median, and mode are all equal: These three measures of central tendency coincide at the center of the curve.
  • Defined shape: Approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

This distribution is common in life sciences, economics, and various kinds of behavioral data, making it a pivotal concept in statistical analysis.
Standard Deviation
The standard deviation is a measure of how spread out numbers are within a data set.
It's essentially the average distance each data point is from the mean.
When examining a normally distributed set of scores, the standard deviation gives us insight into how much variability we can expect.
  • Low standard deviation: Data points are closely clustered around the mean.
  • High standard deviation: Data points are more spread out.

This concept is crucial for interpreting the scores of students in the advanced first aid course.
Knowing the standard deviation helps determine whether a student's score is usual or unusual in the context of the class.
Mean Calculation
Calculating the mean, or average, is simple yet powerful in summarizing a data set.
You add up all the individual scores and then divide by the total number of scores.
In formula terms:\[\text{Mean} \ (\mu) = \frac{\text{Sum of all scores}}{\text{Total number of scores}}\]To determine if a student's score from the advanced first aid course falls above or below average, knowing the mean is essential.
For the given exercise, the mean score was provided as 150, helping to interpret the z-scores by showing each student's performance relative to this average.
Advanced First Aid Course Scores
Understanding the scores from the advanced first aid course involves interpreting them in light of both the mean and standard deviation.
Each student's score can be analyzed using z-scores, which tell us how far each individual score is from the mean in units of standard deviation.
  • Scoring above average: Students like Robert, Juan, and Linda scored higher than the mean.
  • Scoring exactly at the average: Joel's score was precisely average.
  • Scoring below average: Jan and Susan's scores fell below the mean.

This approach generates a clear understanding of who excelled, who met expectations, and who might need extra help, thereby providing a useful snapshot of overall class performance.

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Most popular questions from this chapter

Find the \(z\) value described and sketch the area described.Find \(z\) such that \(8 \%\) of the standard normal curve lies to the right of \(z\).

When we use a normal distribution to approximate a binomial distribution, why do we make a continuity correction?

Watches Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 28 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 5 months, and the distribution of lifetimes is normal. (a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production should the company expect to replace? (b) Inverse Normal Distribution If Accrotime does not want to make refunds on more than \(12 \%\) of the watches it makes, how long should the guarantee period be (to the nearest month)?

Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 45 months and a standard deviation of 8 months. (a) If Quick Start guarantees a full refund on any battery that fails within the 36 -month period after purchase, what percentage of its batteries will the company expect to replace? (b) Inverse Normal Distribution If Quick Start does not want to make refunds for more than \(10 \%\) of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

Find the \(z\) value described and sketch the area described.Find \(z\) such that \(97.5 \%\) of the standard normal curve lies to the left of \(z\).
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