Table of contents
- 1. Intro to Stats and Collecting Data55m
- 2. Describing Data with Tables and Graphs1h 55m
- 3. Describing Data Numerically1h 45m
- 4. Probability2h 16m
- 5. Binomial Distribution & Discrete Random Variables2h 33m
- 6. Normal Distribution and Continuous Random Variables1h 38m
- 7. Sampling Distributions & Confidence Intervals: Mean1h 3m
- 8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- 9. Hypothesis Testing for One Sample1h 1m
- 10. Hypothesis Testing for Two Samples2h 8m
- 11. Correlation48m
- 12. Regression1h 4m
- 13. Chi-Square Tests & Goodness of Fit1h 20m
- 14. ANOVA1h 0m
6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
Problem 6.2.37a
Textbook Question
Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores.
a. If she curves by adding 15 to each grade, what is the new mean and standard deviation?

1
Understand the problem: The professor is curving the test scores by adding 15 to each grade. The original distribution of scores is normally distributed with a mean (μ) of 60 and a standard deviation (σ) of 12. We need to determine the new mean and standard deviation after the adjustment.
Recall the property of normal distributions: When a constant value is added to every data point in a dataset, the mean of the dataset increases by that constant, but the standard deviation remains unchanged. This is because standard deviation measures the spread of the data, which is unaffected by adding a constant.
Calculate the new mean: Add the constant value (15) to the original mean (60). Use the formula: , where is the constant being added.
Determine the new standard deviation: Since adding a constant does not affect the spread of the data, the new standard deviation remains the same as the original standard deviation. Use the formula: .
Summarize the results: The new mean is the original mean plus 15, and the new standard deviation is the same as the original standard deviation. These values describe the adjusted normal distribution of the test scores.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the test scores follow a normal distribution with a mean of 60 and a standard deviation of 12.
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Mean and Standard Deviation
The mean is the average of a set of values, calculated by summing all the values and dividing by the number of values. The standard deviation measures the amount of variation or dispersion in a set of values. In the case of curving test scores, understanding how these two statistics change when a constant is added is crucial for determining the new mean and standard deviation.
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Effects of Linear Transformation
A linear transformation involves adding or multiplying a constant to a dataset. When a constant is added to each score, the mean increases by that constant, while the standard deviation remains unchanged. This principle is essential for calculating the new mean and standard deviation after curving the test scores by adding 15.
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