Probability Distributions

Identifying Probability Distributions

A probability distribution describes how probabilities are distributed over the values of a random variable. For a distribution to be a valid probability distribution:

• Each probability must be between 0 and 1, inclusive.

• The sum of all probabilities must equal 1.

Example: If a random variable X can take values 1, 2, 3 with probabilities 0.2, 0.5, and 0.3, respectively, this is a valid probability distribution because all probabilities are between 0 and 1 and their sum is 1.

Binomial Distributions

A binomial experiment is a statistical experiment that satisfies the following conditions:

• The experiment consists of n repeated trials.

• Each trial has only two possible outcomes: success or failure.

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

• The trials are independent.

The mean and variance of a binomial distribution are given by:

• Mean:

• Variance:

Example: For 10 coin tosses (n = 10) with probability of heads p = 0.5, the mean is and the variance is .

Normal Probability Distributions

Standard Normal Distribution and Z-Scores

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. The z-score measures how many standard deviations a value is from the mean:

• Z-score formula:

To find the area under the standard normal curve (probability), use z-tables or technology.

• To find the area to the left of a z-value, look up the value in the z-table.

• To find the area between two z-values, subtract the smaller area from the larger area.

Example: The area to the left of z = 1.0 is approximately 0.8413.

Finding Probabilities for Normal Distributions

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

• Convert X to a z-score using .

• Use the z-score to find probabilities from the standard normal table.

Example: If , the probability that is found by computing and looking up the area to the left of z = 1.33.

Percentiles and Z-Scores

To find the z-score corresponding to a given percentile, use the standard normal table in reverse. For example, the 90th percentile corresponds to a z-score of approximately 1.28.

Comparing Values Using Z-Scores

To compare values from different normal distributions, convert each value to a z-score and compare the z-scores. The higher z-score indicates a value further above its mean.

Sampling Distributions and the Central Limit Theorem (CLT)

Sampling Distribution of the Mean

The sampling distribution of the sample mean describes the distribution of sample means over repeated sampling from the same population. According to the Central Limit Theorem (CLT):

• The sampling distribution of the mean approaches a normal distribution as the sample size increases (n > 30 is a common rule of thumb).

• Mean of the sampling distribution:

• Standard deviation (standard error):

Confidence Intervals

Margin of Error

The margin of error quantifies the uncertainty in an estimate. For a confidence interval for the mean with known standard deviation:

• Margin of error:

Where is the critical value from the standard normal distribution for the desired confidence level.

Constructing Confidence Intervals

A confidence interval for the population mean (with known ) is:

For unknown , use the t-distribution:

Example: For , , , and 95% confidence ():

Sample Size Determination

To find the required sample size for a given margin of error E, confidence level, and standard deviation:

Hypothesis Testing

Choosing the Appropriate Test

Use the z-test when the population standard deviation is known and the sample size is large (n > 30). Use the t-test when the population standard deviation is unknown and/or the sample size is small (n < 30).

Conditions for Hypothesis Tests

• 1-sample z-test: Population standard deviation known, normal population or large n.

• 1-sample t-test: Population standard deviation unknown, normal population or large n.

• 2-sample z-test: Both population standard deviations known, independent samples.

• 2-sample t-test (independent): Population standard deviations unknown, independent samples.

• 2-sample t-test (dependent/paired): Paired or matched samples.

Type I and Type II Errors

• Type I Error (\(\alpha\)): Rejecting the null hypothesis when it is true.

• Type II Error (\(\beta\)): Failing to reject the null hypothesis when it is false.

Example: In a medical test, a Type I error means concluding a treatment works when it does not; a Type II error means failing to detect a real effect.

One-Tailed vs. Two-Tailed Tests

• One-tailed test: Tests for deviation in one direction (e.g., greater than or less than).

• Two-tailed test: Tests for deviation in either direction (e.g., not equal to).

Rejecting or Failing to Reject the Null Hypothesis

• If the p-value < significance level (\(\alpha\)), reject the null hypothesis.

• If the p-value >= significance level, fail to reject the null hypothesis.

• Alternatively, if the confidence interval does not contain the hypothesized value, reject the null hypothesis.

Critical Values and Rejection Regions

The critical value is the cutoff value that defines the rejection region for the test statistic. For a given significance level, find the corresponding z or t value from statistical tables.

Dependent vs. Independent Samples

• Independent samples: Samples are unrelated (e.g., two different groups).

• Dependent samples: Samples are paired or matched (e.g., before-and-after measurements on the same subjects).

Steps for Hypothesis Testing (One Sample)

1. State the null and alternative hypotheses.

2. Choose the significance level (\(\alpha\)).

3. Calculate the test statistic (z or t).

4. Find the critical value(s) or p-value.

5. Make a decision: reject or fail to reject the null hypothesis.

6. State the conclusion in context.

Summary Table: Key Formulas

Additional info: This guide synthesizes key concepts from probability distributions, normal distributions, sampling, confidence intervals, and hypothesis testing, as outlined in the study guide. It is intended as a comprehensive review for exam preparation.