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Chapter 3: Problem 54

Suppose that your statistics professor returned your first midterm exam with only a \(z\) -score written on it. She also told you that a histogram of the scores was mound shaped and approximately symmetric. How would you interpret each of the following \(z\) -scores? a. 2.2 b. 0.4 c. 1.8 d. 1.0 e. 0

Short Answer

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a. A z-score of 2.2 indicates that the student scored better than approximately 98.2% of the class, as their score was 2.2 standard deviations above the mean. b. A z-score of 0.4 indicates that the student scored better than approximately 65.5% of the class, as their score was 0.4 standard deviations above the mean. c. A z-score of 1.8 indicates that the student scored better than approximately 96.4% of the class, as their score was 1.8 standard deviations above the mean. d. A z-score of 1.0 indicates that the student scored better than approximately 84.1% of the class, as their score was 1 standard deviation above the mean. e. A z-score of 0 indicates that the student scored exactly at the average performance, with about 50% of the class scoring better and 50% scoring worse.

Step by step solution

01

Understanding Normal Distribution and the Z-Score

The z-score, also known as the standard score, represents how many standard deviations an observation is from the mean of the distribution. A positive z-score indicates the observation is above the mean, and a negative z-score indicates the observation is below the mean. In a normal distribution: - Approximately 68% of the observations lie within 1 standard deviation of the mean (z-scores between -1 and 1). - Approximately 95% of the observations lie within 2 standard deviations of the mean (z-scores between -2 and 2). - Nearly all observations lie within 3 standard deviations of the mean (z-scores between -3 and 3).
02

Interpreting the Given Z-Scores

We will interpret each z-score based on the information provided about the normal distribution. a. 2.2 This z-score is above the mean and within 2.2 standard deviations. The student scored better than approximately 98.2% of the class. b. 0.4 This z-score is above the mean and within 0.4 standard deviations. The student scored better than approximately 65.5% of the class. c. 1.8 This z-score is above the mean and within 1.8 standard deviations. The student scored better than approximately 96.4% of the class. d. 1.0 This z-score is above the mean and within 1 standard deviation. The student scored better than approximately 84.1% of the class. e. 0 This z-score is equal to the mean. The student scored exactly at the average performance, with about 50% of the class scoring better and 50% scoring worse. In conclusion, the z-scores help to understand a student's performance relative to the class's average performance. The higher the z-score, the better the student performed compared to their classmates.

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Key Concepts

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

Normal Distribution
The normal distribution, commonly known as the bell curve, is a fundamental concept in statistics that represents how data points are distributed around the mean. It assumes that most observations cluster around the central peak and probabilities for falling further away from the mean decrease exponentially. This pattern results in the symmetric, bell-shaped curve.

Properties of a normal distribution include its symmetry and that it is fully described by two parameters: its mean and its standard deviation. The area under the curve corresponds to the probability of occurrence, which allows statisticians to predict the likelihood of different outcomes.
Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. In the context of the normal distribution, the standard deviation works as a ruler to determine how far an observation is from the mean. It is the distance from the center to the inflection points of the bell curve, where it changes concavity.
Statistical Analysis
Statistical analysis involves collecting, reviewing, interpreting, and presenting data to discover underlying patterns and trends. It can be descriptive or inferential. Descriptive statistics summarize data from a sample using measures such as mean and standard deviation. Inferential statistics, on the other hand, allow predictions to be made from a sample to a population. Z-scores are part of this analytical toolset, enabling comparisons across different sets of data.
Relative Performance Measurement
Relative performance measurement refers to evaluating an individual's or entity's performance in relation to a benchmark or peer group. It's widely used in educational grading, financial market assessments, and employee evaluations. In the exercise provided, z-scores are used to assess a student's performance relative to their classmates. A z-score of 0 would indicate average performance, while a positive z-score signifies above-average performance and a negative z-score below-average performance. It鈥檚 an effective way to see where someone stands in a normally distributed set of data.

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Most popular questions from this chapter

The state of California defines family income groups in terms of median county income as follows: Extremely low income: below \(30 \%\) of county median income Very low income: between \(30 \%\) and \(50 \%\) of county median income Low income: between \(50 \%\) and \(80 \%\) of county median income Moderate income: between \(80 \%\) and \(120 \%\) of county median income For San Luis Obispo County, the median income as of May 24,2016 for single person households was $$\$ 53,950$$ (www.slocounty.ca.gov/Assets/PL/Housing/AHS/AHS.pdf, August \(1,2016,\) retrieved April 19,2017\()\). a. Interpret the value of the median income for a singleperson household in San Luis Obispo County. b. Each of the following statements is incorrect. For each statement, use the given information to explain why it is incorrect. Statement 1: \(30 \%\) of the single-person households in San Luis Obispo County would be classified as extremely low income. Statement 2: More than \(50 \%\) of the single-person households in San Luis Obispo County would be classified as extremely low income or very low income. Statement 3 : There cannot be any single-person households in San Luis Obispo County that would be classified as having an income that was greater than those in the moderate income category.

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