Sometimes the values in your sample are not whole numbers - but rather proportions!

Proportions are numbers between 0 and 1, such as fractions, decimals or percentages. They represent part of a population or sample that shares the same characteristic.

For example: We may think (hypothesize) that 30% of the population has tested positive for COVID.

However, you may sample100 people of which only 20 tested positive for COVID, which is a proportion of .20 or 20%.

You can claim that the population proportion of 30% is actually lower. Using your sample data, you can test this claim:

H (null): p = .30

H (alternative): p < .30

Notice - here our average is a decimal rather than a hole number. Hence, it's a proportion!

Our population proportion (often referred to as just p or p-zero) is always in the hypothesis.

Proportions are expressed successful trials out of total trials.

Sample proportions (aka p-hat or p-1) are denoted as x out of n: successful trials (x) out of (n) total trials (aka sample size).

Our sample proportion value (aka p-hat or p-1) is x/n.

Only the Z table is used for proportions (not T).

One Proportion Z-Test

A one proportion significance test compares one sample proportion with it's population proportion to see if there is a significant change.

The equation for the z-score of a proportion is:

Two-Proportion Z-Test

A two proportion significance test compares two sample proportions to see if they are significantly different. Our population proportion (p0) would be zero - meaning that there is no difference or change between the two sample proportions.

H (null): p1-p2 = 0

H (alternative): p1-p2 > 0

The equation for the z-score measuring the difference between two proportions is:

You may notice this is similar to our original z statistic equation.

(sample mean - population mean) / standard deviation

p^1 - p^2 = sample mean difference

0 = hypothesized population mean difference (p1-p2)

= sample standard deviation (aka standard error)

Standard Deviation aka Standard Error Equation:

The equation to confirm the "normal" conditions for a proportion are as follows:

Confidence Intervals

Same format as all confidence intervals:

Prediction +/- Margin of Error

Gives us a range of likely (possible) values of which the true population mean must lie.

Point Estimate: p^ (sample proportion aka predicted population proportion)

Margin of Error - Includes the standard deviation (aka standard error) adjusted for our confidence level (Z*).

Z*- The more confident we are (bigger Z), the wider the margin of error, and the less accurate the results.

(We can be more confident because we have a larger range of possible values that we are including. Remember the true mean can only be one of them.)

Sample Size - number of subjects in your sample.

- The larger our sample size (bigger n), the smaller the margin of error, and the more accurate the results.

(The smaller range of possible values resulting from the larger sample size increases accuracy)

Example

Less Than

Greater Than

Equal to

Confidence Interval

Relationship to Binomial Distribution? (Successes / Fixed Number of Trials) - One of two options - = approximate .5 probability, assuming no tendency.

If population proportion (population tendency) is not known, we can use .5 to approximate the population proportion value.

Example

Next, let's take a break - and take a look at ANOVA testing.

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