📘 Study Guide: Chapter 4 – Z-Scores & the Normal Distribution

Normal distribution

A bell-shaped, symmetric distribution where mean = median = mode.

Standard normal distribution (Z-distribution)

Normal distribution with mean = 0 and standard deviation = 1.

Empirical Rule (68–95–99.7 Rule)

68% of values lie within ±1 standard deviation of the mean, 95% within ±2 SD, and 99.7% within ±3 SD.

Z-score

The number of standard deviations a data point is from the mean.

Central Limit Theorem (CLT)

States that with a sample size greater than 30, the sampling distribution of the mean approximates a normal distribution.

Probability (P)

The likelihood of an event occurring, represented as the area under the curve.

Z-table

A table that converts z-scores to probabilities, indicating the area under the normal distribution curve.

Population Z-score

The calculation of the z-score using population parameters: z = (x - μ) / σ.

Sample Z-score

The calculation of the z-score using sample parameters: z = (x - x̄) / s.

Convert Z back to Raw Score

Formula for converting z back to raw score: x = zσ + μ.

P(x<a)

Probability value that is less than a.

P(x>a)

Probability value that is greater than a.

P(a<x<b)

Probability value that is between a and b.

Comparing Scores using Z-scores

Z-scores can be used to compare values from different distributions, like ACT vs. SAT scores.

CLT approximation

If the data is not normal but the sample size is greater than 30, the Central Limit Theorem allows for z-approximation.

Calculating probabilities steps

1. Convert raw values to z-scores. 2. Use the z-table. 3. Interpret results as the area under the curve.

ACT score example

For an ACT score of 23 with μ=19 and σ=4, z = 1.

SAT score example

For a SAT score of 1140 with μ=1100 and σ=400, z = 0.1.

Finding raw score from Z

To find the raw SAT score for z=0.5 with μ=1100 and σ=200, raw score x = (0.5)(200)+1100 = 1200.

Soda price example

For soda prices with μ=2.15 and σ=0.15, P(x<2.11) is found by converting to z and using the z-table.

Interval probability example

For values 2.00<x<2.30 with μ=2.15 and σ=0.15, the probability equals 0.68 (68%).

GRE score z-score calculation

For GRE score of 162 with μ=150 and σ=8, z = (162 - 150) / 8.

Converting z to raw score

If z = -1.5, convert to raw score using μ=100 and σ=15.

P(x>70) for normal distribution

Calculate for a normal distribution with μ=60 and σ=10.

Percentages of students scoring 900 to 1300

Determine for SAT score dataset with mean = 1100 and σ = 200.

Finding corresponding z-score

If P(z<a) = 0.80, find the z-score associated with this probability.

Z-distribution characteristic

A z-distribution is a type of normal distribution where mean = 0 and standard deviation = 1.

68% rule interpretation

According to the empirical rule, 68% of data falls within one standard deviation of the mean.

Mean, Median, Mode in Normal Distribution

In a normal distribution, mean = median = mode.

Effect of sample size on distribution

With a larger sample size, the sampling distribution of the mean approaches a normal shape according to the CLT.

Area under normal curve

The total area under a normal curve is always equal to 1.00.

Interpretation of positive z-score

A positive z-score indicates that the value is above the mean.

Negative values in z-scores

Z-scores can be negative, while standard deviations and variance cannot.

Standard normal distribution parameters

In a standard normal distribution, μ = 0 and σ = 1.

Purpose of z-scores

Z-scores are used to standardize scores from different distributions for comparison.

Lower tail probability interpretation

If P(x<a)=0.30, then a is below the mean.

Finding probabilities using z-table

Use the z-table to find the area under the normal curve for specific z-scores.

Z-score and standard deviation relationship

A z-score describes how many standard deviations a data point is above or below the mean.

Calculating probability of P(x<a)

Calculate using the normal distribution parameters for the event of interest.

Calculation of P(x>a)

Determine what percentage of values are greater than a specific score in a distribution.

Probability less than a specified value

Use z-scores to find probabilities of observations below a particular value.

Z-score for specific data point

Calculate the z-score to understand its position relative to the distribution.

Values within standard deviations

The empirical rule allows estimation of percentage of data falling within one, two or three standard deviations.

Probability calculations for intervals

To find probabilities for intervals, determine z-scores for both endpoints.

Finding area under the curve for multiple regions

Use z-scores to find cumulative probabilities straddling different values.

Using distributions in applications

Distributions can be useful in determining probabilities and making comparisons across different datasets.

Understanding the implications of the z-score

A z-score provides insight into how extreme or normal a data point is within its distribution.

Converting raw scores to standardized scores

Use the z-score formulas to transform raw score data for analysis.

Application of the Central Limit Theorem in research

The CLT is pivotal in simplifying the analysis of sample means in inferential statistics.

Direct interpretation of z-tables

Z-tables allow for quick reference to find probability values corresponding to specific z-scores.

Statistical inference using z-distribution

Utilize z-distributions for making inferences about population parameters from sample data.

Understanding the shape of the distribution

A normal distribution is characterized by its distinct bell shape and symmetry around the mean.

Calculating probabilities using empirical data

Real-world data can often be approximated using normal distributions for relating probabilities.