The Normal Distribution, Revisited - AP Statistics Study Guide

Quick Summary

This guide will empower you to calculate probabilities for any continuous random variable that follows a Normal distribution. You will master the crucial skill of standardizing an observation using a z-score, which allows you to find the probability of observing a value within a specific range. By the end, you will be able to confidently solve Normal distribution problems using a structured, four-step process that guarantees full credit on the AP exam.

Key Concepts

• Continuous Random Variables: Unlike discrete variables that can be counted (e.g., number of heads in 3 coin flips), a continuous random variable can take on any value within a given interval. Examples include height, weight, and time.

• Density Curves: The probability distribution of a continuous random variable is shown by a density curve. The total area under any density curve is exactly 1. The probability of an event is the area under the curve for the corresponding interval of values. A key fact is that the probability of a continuous random variable taking on any single specific value is 0, because the area at a single point is zero.

• The Normal Distribution: This is the most important density curve in statistics. It is a symmetric, single-peaked, bell-shaped curve that is completely described by its mean (μ), which locates the center, and its standard deviation (σ), which measures the spread. We use the notation N(μ, σ) to describe a Normal distribution with a specific mean and standard deviation.

  [Image: A bell-shaped Normal curve. The peak is labeled with μ. The points one, two, and three standard deviations away from the mean (μ+σ, μ+2σ, μ-σ, etc.) are marked on the horizontal axis, showing how the curve spreads out.]

• The Empirical Rule (68-95-99.7 Rule): This rule is a useful approximation for all Normal distributions:

  • Approximately 68% of observations fall within 1 standard deviation of the mean (μ ± σ).

  • Approximately 95% of observations fall within 2 standard deviations of the mean (μ ± 2σ).

  • Approximately 99.7% of observations fall within 3 standard deviations of the mean (μ ± 3σ).

• The Standard Normal Distribution: This is a special Normal distribution with a mean of μ = 0 and a standard deviation of σ = 1. It is denoted as N(0, 1). This is our reference distribution; we convert all other Normal distributions into this standard form to make calculations universal.

• Z-Scores (Standardizing): A z-score measures how many standard deviations an observation $x$ is from the mean. It is the bridge that connects any Normal distribution, N(μ, σ), to the Standard Normal Distribution, N(0, 1). The formula is:

  $z=(x-\mu )/\sigma$

  • A positive z-score means the observation is above the mean.

  • A negative z-score means the observation is below the mean.

  • The process of calculating a z-score is called standardizing.

• The Two-Step Process for Finding Probabilities:

  1. Standardize: Take your value of interest, $x$, from your N(μ, σ) distribution and convert it to a z-score using the formula.

  2. Find Probability: Use the z-score and a calculator (or z-table) to find the corresponding area under the Standard Normal Curve. This area is your probability.

Key Vocabulary

• Normal Distribution: A continuous probability distribution that is symmetric, bell-shaped, and completely described by its mean (μ) and standard deviation (σ).

• Standard Normal Distribution: A specific Normal distribution with a mean of 0 and a standard deviation of 1.

• Z-score: A standardized value that indicates the number of standard deviations an observation is from the mean.

• Standardize: The process of converting an observation $x$ from a Normal distribution N(μ, σ) into a z-score.

• Density Curve: A curve that is always on or above the horizontal axis and has a total area of exactly 1 underneath it. It describes the overall pattern of a distribution.

• Percentile: The value below which a given percentage of observations in a distribution falls. For example, the 90th percentile is the value that is greater than 90% of all other values.

Calculator Tech (TI-84)

Two functions are essential for this topic. They are found under [2nd] -> [VARS] (DISTR).

1. $normalcdf(lower,upper,\mu ,\sigma )$: Calculates the area (probability) under a Normal curve between a $lower$ bound and an $upper$ bound.

   • To find P(X < a), use lower: -1E99 (a very small number) and upper: a.

   • To find P(X > b), use lower: b and upper: 1E99 (a very large number).

   • To find P(a < X < b), use lower: a and upper: b.

2. $invNorm(area,\mu ,\sigma )$: Calculates the boundary value $x$ given the area to its left. This is used to find percentiles.

   • Crucial Note: The $area$ input must always be the cumulative area from the far left of the distribution. To find the value for the top 10%, you must use an $area$ of 0.90.

How to Show Work on the FRQ

To earn full credit on a Normal distribution FRQ, you must clearly communicate your process. Use the following four-step model. Simply writing the calculator command and the answer is not enough.

The 4-Step Process for Normal Distribution Problems:

1. State: Define the random variable and state the distribution with its parameters (mean and standard deviation). Draw and label a Normal curve.

   • Template: "Let X = [describe the variable in context]. X follows a Normal distribution with a mean of μ = [value] and a standard deviation of σ = [value]. We can write this as N(μ, σ)."

   • Action: Draw a bell curve, label the mean in the center, and mark the value(s) of interest. Shade the area you are trying to find.

2. Plan: State the probability you are trying to find and show the z-score calculation for each boundary value.

   • Template: "We want to find P([inequality statement, e.g., X > 72]). The z-score for the boundary value is: z = (x - μ) / σ = ([value] - [mean]) / [std dev] = [z-score]."

3. Do: Show the calculator command you used and the resulting probability.

   • Template: "Using technology: normalcdf(lower: [value], upper: [value], μ: [value], σ: [value]) = [probability]."

4. Conclude: Write a clear sentence that answers the question in the context of the problem.

   • Template: "There is a [probability] chance that a randomly selected [subject] will have a [variable] of [less than/greater than/between] [value(s)]."

Practice Problems

Problem 1: The scores on a recent nationwide standardized test are approximately Normally distributed with a mean of 1050 and a standard deviation of 200. What is the probability that a randomly selected student scores between 1000 and 1300?

Solution:

State: Let X be the score of a randomly selected student on the standardized test. The scores follow a Normal distribution with μ = 1050 and σ = 200, or N(1050, 200). We want to find the probability that a student scores between 1000 and 1300.

[Image: A Normal curve centered at 1050. The values 1000 and 1300 are marked on the x-axis, and the area between them is shaded.]

Plan: We want to find P(1000 < X < 1300). First, we find the z-scores for the boundaries.

For x = 1000: z = (1000 - 1050) / 200 = -0.25

For x = 1300: z = (1300 - 1050) / 200 = 1.25

Do: Using technology: normalcdf(lower: 1000, upper: 1300, μ: 1050, σ: 200) = 0.492.

Conclude: There is a 0.492 probability that a randomly selected student will have a test score between 1000 and 1300.

Problem 2: The lengths of a certain species of fish are Normally distributed with a mean of 25 cm and a standard deviation of 4 cm. To be kept, a fish must be in the top 15% by length. What is the minimum length a fish must be to be kept?

Solution:

State: Let L be the length of a randomly selected fish. The lengths follow a Normal distribution with μ = 25 and σ = 4, or N(25, 4). We need to find the length that separates the top 15% of fish from the bottom 85%.

[Image: A Normal curve centered at 25. A vertical line is drawn towards the right tail, with the area to its right shaded and labeled "0.15". The area to the left is labeled "0.85". The boundary value is marked with a "?".]

Plan: We are looking for a length, $k$, such that P(L > k) = 0.15. This is equivalent to finding the 85th percentile, since the calculator's $invNorm$ function requires the area to the left.

Do: We use the inverse normal function with an area of 1 - 0.15 = 0.85 to the left.

Using technology: invNorm(area: 0.85, μ: 25, σ: 4) = 29.146 cm.

Conclude: A fish must be at least 29.146 cm long to be kept.

Common Mistakes to Avoid

• $invNorm$ Area Error: The most common mistake is forgetting that $invNorm$ requires the area to the left of the value you're looking for. If you need to find the cutoff for the top 5% (the 95th percentile), you must input area: 0.95, not area: 0.05.

• Confusing Z-score with Probability: A z-score is not a probability. A z-score tells you a location on the horizontal axis (how many standard deviations from the mean), while a probability is an area under the curve (always between 0 and 1). Z-scores can be negative, but probabilities cannot.

• Forgetting Context: Your final answer must be a sentence that interprets the numerical result in the context of the problem. Simply writing "0.0668" is not a complete answer on an FRQ.

• Calculator Input Errors: Be very careful with the $lower$ and $upper$ bounds in $normalcdf$. For a "greater than" probability like P(X > 100), the $lower$ bound is 100 and the $upper$ bound is a very large number like 1E99 (or 10^99). Using -1E99 by mistake will give you the wrong area.

• Not Showing the Z-score Calculation: Even when using $normalcdf$ with the mean and standard deviation directly, you must still show the z-score formula and calculation as part of your "Plan" step on an FRQ to demonstrate a full understanding of the standardization process.