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Chapter 3: Problem 50

Use the five test scores of 65,70 \(71,75,\) and 95 to answer the following questions: (a) Find the sample mean. (b) Find the median. (c) Which measure of central tendency best describes the typical test score? (d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores. (e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

Short Answer

Expert verified
a) 75.2 b) 71 c) Median d) 79.2 e) The mean increased by 4, from 75.2 to 79.2.

Step by step solution

01

List the Given Scores

The given test scores are 65, 70, 71, 75, and 95.
02

Calculate the Sample Mean

The sample mean is calculated by summing all the test scores and dividing by the number of scores. Compute the sum: \(65 + 70 + 71 + 75 + 95 = 376\). Divide the sum by the number of scores: \ \ \text{Sample Mean} = \frac{376}{5} = 75.2
03

Find the Median

To find the median, list the scores in ascending order: 65, 70, 71, 75, 95. The median is the middle value, which in this case is 71.
04

Determine the Best Measure of Central Tendency

Compare the sample mean (75.2) and the median (71). The median (71) is closer to the typical score and less influenced by the extreme value (95), making it a better measure of central tendency in this case.
05

Adjust Scores by Adding 4 Points

Add 4 points to each test score: 65 + 4 = 69, 70 + 4 = 74, 71 + 4 = 75, 75 + 4 = 79, 95 + 4 = 99. The adjusted scores are 69, 74, 75, 79, and 99.
06

Compute the Adjusted Sample Mean

Sum all the adjusted scores: \(69 + 74 + 75 + 79 + 99 = 396\). Divide the sum by the number of scores: \ \ \text{Adjusted Sample Mean} = \frac{396}{5} = 79.2
07

Compare Unadjusted and Adjusted Means

The unadjusted sample mean is 75.2, while the adjusted sample mean is 79.2. Adding 4 points to each score increased the mean by 4 points, from 75.2 to 79.2.

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Key Concepts

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

Sample Mean
The sample mean represents the average of a set of values. It's calculated by summing all the values in the sample and then dividing by the number of values. In the given exercise, the test scores were 65, 70, 71, 75, and 95.
To compute the sample mean (denoted as \(\bar{x}\)), first sum all the test scores: \(65 + 70 + 71 + 75 + 95 = 376\).Then divide this sum by the number of scores, in this case, 5: \(\bar{x} = \frac{376}{5} = 75.2\).Hence, the sample mean is 75.2. It's a vital measure in statistics because it gives a central value for the dataset.
Remember, mean can be influenced by very high or low values, also known as outliers.
Median
The median is another measure of central tendency. It indicates the middle value in a dataset when the numbers are arranged in ascending order. Unlike the mean, the median is not affected by outliers.
To find the median from the given scores (65, 70, 71, 75, 95), list them in ascending order if they aren't already: 65, 70, 71, 75, 95.
Since there is an odd number of scores (5), the median is the middle number, which is 71. If there were an even number of scores, the median would be the average of the two middle numbers.
The median is useful in datasets with skewed distributions because it can better represent the central location of the data.
Curved Scores
In educational settings, curving scores means adjusting the scores to improve results or fit a desired distribution. In the given exercise, the professor decided to curve the exam scores by adding 4 points to each test score.
The original scores were: 65, 70, 71, 75, 95. After adding 4 points to each score, the adjusted scores became: 69, 74, 75, 79, 99.
Curving can help increase the average score and ensure fairness if the exam was unusually difficult. It is a common practice used to standardize scores across different test editions or to correct for difficult test questions.
Statistical Analysis
Statistical analysis involves collecting, summarizing, and interpreting data to discover patterns and trends. It helps answer questions like 'What is the typical test score?' or 'How does changing one variable affect outcomes?'
By analyzing the test scores using central tendency measures like the mean and median, we determine the average performance and the most typical score. Comparing the unadjusted and adjusted (curved) means helps us understand the effect of curving.
Here, the unadjusted mean was 75.2, and after curving, it increased to 79.2. This shows the added points simply shift the distribution without affecting the relative performance of each student.

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Most popular questions from this chapter

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