The Normal Distribution, Revisited

#The Normal Distribution: Your Ultimate Guide 🚀

Welcome to your final review of the normal distribution! This guide is designed to be your go-to resource, helping you feel confident and fully prepared for the AP Statistics exam. Let's dive in!

#What is a Normal Distribution?

The normal distribution, often called the "bell curve," is a fundamental concept in statistics. It's a continuous probability distribution that's:

• Symmetrical: The left and right sides are mirror images. 👯
• Unimodal: It has a single peak or mound. 🐪
• Mean = Median = Mode: The center of the distribution is where all three measures of central tendency meet.

#Describing a Normal Model

To fully describe a normal model, consider these three key aspects:

• Center: The mean (μ), which is the arithmetic average of the data.
• Shape: The symmetrical bell curve.
• Spread: The standard deviation (σ), which measures the dispersion of data around the mean.

#Conditions for Normality

Before you can assume a normal model, you need to check specific conditions:

#The Empirical Rule (68-95-99.7 Rule)

The empirical rule is a quick way to estimate the proportion of data within a certain number of standard deviations from the mean:

• 68% of the data falls within 1 standard deviation of the mean. 🔴
• 95% of the data falls within 2 standard deviations of the mean. 🟣
• 99.7% of the data falls within 3 standard deviations of the mean. 🟢

#Source: Medium

#Z-Scores: Standardizing Data

A z-score tells you how many standard deviations a data point is from the mean. It's calculated as:

$$z = \frac{x - \mu}{\sigma}$$

Where:

• x is the data point

• μ is the mean

• σ is the standard deviation

• Positive z-score: The data point is above the mean.

• Negative z-score: The data point is below the mean.

#Interpreting Percentiles in Normal Distributions

Percentiles tell you the percentage of data that falls below a certain value. In a normal distribution, you can find the z-score corresponding to a given percentile using a z-table or calculator. For example:

• The 95th percentile corresponds to the z-score where 95% of the data falls below it.

#Source: College Board

#Practice Problems

Let's solidify your understanding with some practice questions:

(1) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a standard deviation of 3 inches. Calculate the z-score for a man who is 72 inches tall.

(2) A study is conducted to compare the GPA of students at two different colleges. The mean GPA of students at College A is 3.0, with a standard deviation of 0.5. Calculate the z-score for a student at College A who has a GPA of 3.5. (3) A study is conducted to compare the number of hours spent studying per week by students at two different universities. The mean number of hours spent studying per week at University A is 15 hours, with a standard deviation of 3 hours. Calculate the z-score for a student at University A who studies 21 hours per week.

(4) A study is conducted to compare the heights of men and women in a certain population. The mean height of men is 70 inches, with a standard deviation of 3 inches. Calculate the z-score for a woman who is 62 inches tall. The mean height of women is 65 inches, with a standard deviation of 2.5 inches.

(5) Going back to problem (2), calculate the z-score for a student at College B who has a GPA of 3.0. The mean GPA of students at College B is 2.5, with a standard deviation of 0.4. ### Solutions

(1) z = (72 - 70) / 3 = 0.67

(2) z = (3.5 - 3.0) / 0.5 = 1

(3) z = (21 - 15) / 3 = 2

(4) z = (62 - 65) / 2.5 = -1.2

(5) z = (3.0 - 2.5) / 0.4 = 1.25

Practice Question

Multiple Choice Questions:

1. The distribution of weights of a certain species of bird is approximately normal with a mean of 150 grams and a standard deviation of 20 grams. What percentage of birds weigh between 130 and 170 grams? (a) 34% (b) 68% (c) 95% (d) 99.7% (e) Cannot be determined

2. A standardized test has a normal distribution with a mean of 500 and a standard deviation of 100. A student scores 650 on the test. What is the z-score for this student's score? (a) -1.5 (b) -0.67 (c) 0.67 (d) 1.5 (e) 2.5

Free Response Question:

The scores on a statistics exam are normally distributed with a mean of 75 and a standard deviation of 8. (a) What is the probability that a randomly selected student scored above 90? (b) What is the probability that a randomly selected student scored between 70 and 80? (c) If 500 students took the exam, approximately how many students scored above 85?

Scoring Guide:

(a) Probability of scoring above 90:

• Calculate the z-score: z = (90 - 75) / 8 = 1.875 (1 point)
• Find the probability using a z-table or calculator: P(Z > 1.875) ≈ 0.0304 (1 point)
• State the probability: There is approximately a 3.04% chance that a randomly selected student scored above 90. (1 point)

(b) Probability of scoring between 70 and 80:

• Calculate the z-score for 70: z1 = (70 - 75) / 8 = -0.625 (1 point)
• Calculate the z-score for 80: z2 = (80 - 75) / 8 = 0.625 (1 point)
• Find the probability using a z-table or calculator: P(-0.625 < Z < 0.625) ≈ 0.468 (1 point)
• State the probability: There is approximately a 46.8% chance that a randomly selected student scored between 70 and 80. (1 point)

(c) Number of students scoring above 85:

• Calculate the z-score: z = (85 - 75) / 8 = 1.25 (1 point)
• Find the probability using a z-table or calculator: P(Z > 1.25) ≈ 0.1056 (1 point)
• Multiply the probability by the number of students: 0.1056 * 500 ≈ 52.8 (1 point)
• State the approximate number of students: Approximately 53 students scored above 85. (1 point)

#Final Exam Focus

Here's a quick rundown of what to focus on for the exam:

• Understanding the Normal Distribution: Know its properties (symmetry, unimodal), and how to describe it using mean and standard deviation.
• Checking Conditions: Always verify the conditions for normality before applying normal models.
• Empirical Rule: Be able to quickly estimate percentages within 1, 2, and 3 standard deviations.
• Z-Scores: Master calculating and interpreting z-scores. Remember, direction matters!
• Percentiles: Understand how to find z-scores corresponding to given percentiles using z-tables or calculators.

Normal distributions are a cornerstone of AP Statistics. Expect to see them in multiple contexts, often combined with other topics like sampling distributions and hypothesis testing.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
• Show Your Work: Even if you get the wrong answer, you can earn partial credit for showing your process.
• Units: Always include units in your answers where applicable.
• Context: Interpret your results in the context of the problem. What do the numbers mean in the real world?
• Practice, Practice, Practice: The more you practice, the more comfortable you'll feel on the exam.

You've got this! Go into the exam with confidence, and remember all you've learned. Good luck! 🍀