Normal Distributions Worksheet 12 7 Answers

Normal Distributions Worksheet: A Comprehensive Guide to Solving Problems

This worksheet tackles problems related to the normal distribution, a fundamental concept in statistics. Understanding normal distributions is crucial for various fields, including finance, engineering, and healthcare. This guide will provide detailed explanations and solutions for common problems encountered in such worksheets. We'll explore different scenarios and techniques to help you master this important topic.

Understanding the Normal Distribution

Before diving into the problems, let's briefly review the key characteristics of a normal distribution. It's also known as a Gaussian distribution, characterized by its bell-shaped curve. Key features include:

• Symmetry: The distribution is perfectly symmetrical around its mean (average).
• Mean, Median, and Mode: The mean, median, and mode are all equal and located at the center of the distribution.
• Standard Deviation: This measures the spread or dispersion of the data. A larger standard deviation indicates a wider spread, while a smaller standard deviation indicates a narrower spread.
• Empirical Rule (68-95-99.7 Rule): Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

These properties are crucial for solving problems related to normal distributions.

Problem Solving Techniques

We’ll primarily utilize z-scores to solve problems related to the normal distribution. A z-score represents the number of standard deviations a data point is away from the mean. The formula for calculating a z-score is:

z = (x - μ) / σ

Where:

• z is the z-score
• x is the data point
• μ is the population mean
• σ is the population standard deviation

Once you have the z-score, you can use a z-table (or statistical software) to find the probability associated with that z-score. This probability represents the area under the normal curve to the left of the z-score.

Example Problems and Solutions

Let's work through some example problems that commonly appear in normal distribution worksheets:

Problem 1: Finding Probabilities

• Scenario: The heights of adult women in a certain population are normally distributed with a mean of 65 inches and a standard deviation of 2.5 inches. What is the probability that a randomly selected woman is taller than 68 inches?

• Solution:

1. Calculate the z-score: z = (68 - 65) / 2.5 = 1.2
2. Look up the z-score in a z-table: A z-score of 1.2 corresponds to a probability of approximately 0.8849. This represents the probability that a woman is shorter than 68 inches.
3. Find the probability of being taller: Since we want the probability of being taller than 68 inches, we subtract the probability from 1: 1 - 0.8849 = 0.1151.

Therefore, the probability that a randomly selected woman is taller than 68 inches is approximately 0.1151 or 11.51%.

Problem 2: Finding Values from Probabilities

• Scenario: The scores on a standardized test are normally distributed with a mean of 500 and a standard deviation of 100. What score separates the top 10% of students from the rest?

• Solution:

1. Find the z-score corresponding to the top 10%: This means finding the z-score that corresponds to a probability of 0.90 (1 - 0.10) in a z-table. The z-score is approximately 1.28.
2. Use the z-score formula to solve for x: 1.28 = (x - 500) / 100. Solving for x gives x = 628.

Therefore, a score of 628 separates the top 10% of students from the rest.

Problem 3: Working with Percentiles

• Scenario: The weights of newborn babies are normally distributed with a mean of 7.5 pounds and a standard deviation of 1.2 pounds. What is the weight of a baby at the 90th percentile?

• Solution:

1. Find the z-score corresponding to the 90th percentile: This is the z-score corresponding to a probability of 0.90 in the z-table, which is approximately 1.28.
2. Use the z-score formula to solve for x: 1.28 = (x - 7.5) / 1.2. Solving for x gives x ≈ 9.06.

Therefore, the weight of a baby at the 90th percentile is approximately 9.06 pounds.

Problem 4: Working with Two Values

• Scenario: The lifespan of a certain type of lightbulb is normally distributed with a mean of 1000 hours and a standard deviation of 100 hours. What percentage of lightbulbs have a lifespan between 900 and 1100 hours?

• Solution:

1. Calculate the z-scores:
   • For 900 hours: z = (900 - 1000) / 100 = -1
   • For 1100 hours: z = (1100 - 1000) / 100 = 1
2. Find the probabilities:
   • For z = -1, the probability is approximately 0.1587.
   • For z = 1, the probability is approximately 0.8413.
3. Find the difference: Subtract the probability for z = -1 from the probability for z = 1: 0.8413 - 0.1587 = 0.6826.

Therefore, approximately 68.26% of lightbulbs have a lifespan between 900 and 1100 hours. This aligns with the Empirical Rule.

Problem 5: Sampling Distributions

• Scenario: A large population has a mean IQ of 100 and a standard deviation of 15. If we take a sample of 25 people, what is the probability that the sample mean IQ is less than 95?

• Solution: This problem involves the sampling distribution of the mean. The standard deviation of the sampling distribution (standard error) is calculated as: σ/√n = 15/√25 = 3. The z-score is calculated using the sample mean: z = (95 - 100) / 3 = -1.67. Looking this up in a z-table gives a probability of approximately 0.0475.

Therefore, there is a 4.75% probability that the sample mean IQ is less than 95.

Advanced Concepts and Applications

These examples cover fundamental problems. More advanced worksheets might introduce concepts like:

• Confidence Intervals: Estimating the range within which the true population mean likely lies.
• Hypothesis Testing: Testing claims about population parameters using sample data.
• Non-Normal Distributions: Approximating distributions using transformations or other techniques.
• Central Limit Theorem: Understanding how the sampling distribution of the mean approaches normality as sample size increases.

Tips for Mastering Normal Distributions

• Practice Regularly: The more problems you solve, the better your understanding will become.
• Use a Z-table: Become proficient in using a z-table or statistical software to find probabilities and z-scores.
• Understand the Concepts: Don't just memorize formulas; understand the underlying principles of the normal distribution.
• Seek Help When Needed: If you're struggling with a particular problem, don't hesitate to seek help from a teacher, tutor, or online resources.

By consistently practicing and applying these techniques, you will develop a strong understanding of normal distributions and effectively solve a wide array of problems. Remember that understanding the core concepts, combined with diligent practice, is the key to success in this area of statistics. This thorough understanding will prove invaluable in many fields requiring statistical analysis.