Probability Distributions

Identifying Probability Distributions

A probability distribution describes how probabilities are distributed over the values of a random variable. To determine if a distribution is a probability distribution:

• All probabilities must be between 0 and 1.

• The sum of all probabilities must equal 1.

• Each outcome must be mutually exclusive.

Example: For a random variable X with possible values 1, 2, 3 and probabilities 0.2, 0.5, 0.3, check that 0.2 + 0.5 + 0.3 = 1.

Binomial Distribution

Mean and Variance of Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Mean:

• Variance:

Example: For n = 10 trials and p = 0.3, mean is , variance is .

Identifying Binomial Experiments

An experiment is binomial if:

• There are a fixed number of trials (n).

• Each trial has two possible outcomes (success or failure).

• The probability of success (p) is constant for each trial.

• Trials are independent.

Example: Flipping a coin 5 times and counting heads.

Normal Distribution

Standard Normal Distribution and Areas

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Areas under the curve represent probabilities.

• To find the area to the left of a z-score, use standard normal tables.

• To find the area between two z-scores, subtract the area to the left of the lower z from the area to the left of the higher z.

Example: Area between z = -1 and z = 1 is approximately 0.6826.

Finding Probabilities for Normal Random Variables

For a normal random variable X with mean and standard deviation :

• Convert X to a z-score:

• Use z-tables to find probabilities.

Example: If , , find probability X < 60: .

Percentiles and Z-Scores

The z-score corresponding to a percentile is found using z-tables or inverse normal functions.

• Percentile is the area to the left of the z-score.

• Find z for a given percentile using tables or calculator.

Example: The 90th percentile corresponds to z ≈ 1.28.

Comparing Values Using Z-Scores

Z-scores allow comparison of values from different distributions:

• Higher z-score = value is further above the mean.

• Lower z-score = value is further below the mean.

Example: Compare test scores from different classes by converting to z-scores.

Sampling Distributions and Central Limit Theorem (CLT)

Mean and Standard Deviation of Sampling Distribution

The sampling distribution of the sample mean has:

• Mean:

• Standard deviation:

Example: If , , , then .

Finding Probabilities for Samples (z and t statistics)

Use z or t statistics to find probabilities for sample means:

• Use z if population standard deviation is known and sample size is large.

• Use t if population standard deviation is unknown and sample size is small.

Confidence Intervals

Margin of Error

The margin of error for a confidence interval is:

• (for known )

• (for unknown )

Example: For , , , .

Confidence Interval for Population Mean

A confidence interval estimates the population mean:

• For known :

• For unknown :

Example: , , , , interval is .

Sample Size Calculation

To find the required sample size for a given margin of error:

Example: , , , (round up to 97).

Hypothesis Testing

Normal vs. t Distribution

Choose the appropriate distribution:

• Normal (z): Population standard deviation known, large sample size.

• t: Population standard deviation unknown, small sample size.

Conditions for Hypothesis Tests

Each test has specific conditions:

• 1-sample z: Random sample, normal population or n > 30, known.

• 1-sample t: Random sample, normal population or n > 30, unknown.

• 2-sample z: Two independent samples, known.

• 2 independent t: Two independent samples, unknown.

• 2 dependent t: Paired samples.

Type I and Type II Errors

Errors in hypothesis testing:

• Type I error: Rejecting a true null hypothesis (false positive).

• Type II error: Failing to reject a false null hypothesis (false negative).

Example: Type I: Concluding a drug works when it does not. Type II: Concluding a drug does not work when it does.

One-Tailed vs. Two-Tailed Tests

Determine the direction of the test:

• One-tailed: Tests for an effect in one direction (e.g., greater than).

• Two-tailed: Tests for an effect in both directions (e.g., not equal).

Example: , is two-tailed.

Rejecting or Failing to Reject the Null Hypothesis

Decisions are based on p-value and significance level ():

• If p-value < , reject .

• If p-value > , fail to reject .

Alternatively, use confidence intervals:

• If the confidence interval does not contain the hypothesized value, reject .

Critical Values and Rejection Regions

Critical values define the boundaries of the rejection region:

• For z-tests, use or .

• For t-tests, use or .

Example: For , two-tailed z-test, critical values are .

Dependent vs. Independent Samples

Samples are:

• Independent: No relationship between samples.

• Dependent: Samples are paired or matched (e.g., before and after measurements).

Testing Claims: One Sample Hypothesis Test

Steps for a one-sample hypothesis test:

1. State hypotheses (, ).

2. Choose significance level ().

3. Calculate test statistic (z or t).

4. Find p-value or compare to critical value.

5. Make decision: reject or fail to reject .

6. Interpret the result in context.

Example: Test if the mean weight is 150 lbs using sample data.

Summary Table: Hypothesis Test Types

Additional info: This table summarizes the main types of hypothesis tests covered in the study guide.