BIT2405 ch7-9

Continuous Probability Distribution

For a discrete variable, you can find the probability of one exact value.

Example: P(X=3)

But for a continuous variable, there are infinitely many possible values, so the probability of one exact value is basically zero.

Example:

If X is height, then: P(X=68.0000000...)

is essentially 0 because there are infinitely many possible heights.

So instead, we find the probability that X falls within a range: P(a ≤ X ≤ b)

Normal Distribution Word Problems

For normal word problems, follow these steps:

1. Make sure the problem says the data are normal or bell-shaped.

2. Identify the mean μ\muμ.

3. Identify the standard deviation σ\sigmaσ.

4. Write the probability statement.

5. Convert xxx values into z-scores.

6. Use the z-table.

7. Subtract or use complement if needed.

z-score formula

A z-score tells you how many standard deviations a value is away from the mean.

When Can You Use Normal Approximation to approximate the binomial distribution?

When np ≥ 5 and nq ≥ 5

Mean and Variance for Normal Approximation

Standard Error

The standard error is the standard deviation of a sample statistic.

Simple meaning:

Standard error tells you how much a sample statistic usually varies from sample to sample.

Parameter

A parameter is a number that describes a population.

Parameters usually use Greek letters.

Parameter	Meaning
μ	Population mean
p	Population proportion
σ	Population standard deviation

Statistic

A statistic is a number calculated from a sample.

Statistic	Meaning
x̄	Sample mean
p̂	Sample proportion
s	Sample standard deviation

Central Limit Theorem

It says: If the sample size is large enough, the sampling distribution of the sample mean will be approximately normal, even if the original population is not normal.

Simple version:

Even if the population is weird-shaped, the sample means become bell-shaped when n is large.

Rule of thumb: If the sample size is at least 30 (n ≥ 30), the Central Limit Theorem usually applies.

Sampling Distribution of the Mean important facts

Mean and Standard Error of Sample Proportion

The mean of the sample proportion is population proportion