# One-sample Hypothesis Test for p

Large Sample Proportion Hypothesis Testing

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• In order to conduct a one-sample proportion z-test, the following conditions should be met:

1. The data are a simple random sample from the population of interest.
2. The population is at least 10 times as large as the sample.
3. $n \cdot p \ge 10$ and $n \cdot \left(1 - p\right) \ge 10$, where $n$ is the sample size and $p$ is the true population proportion.
• A one sample z test for a proportion is a test meant to evaluate if a certain proportion $p$ is equal or not to a certain value ${p}_{0}$.
The test is calculated in this way:

z_0 = (hat p - p_0)/(sqrt((p_0*(1-p_0))/n)

where $\hat{p}$ is the proportion calculated on your sample.
then, depending on which type of alternative hypotesis and value of $\alpha$ do you have, you accept or reject the null hypothesis depending on the value of ${z}_{0}$.

• $\setminus \hat{p}$ : the estimated value of the parameter $p$ based on your sample.

$\setminus {p}_{0}$ : the real value of $p$ under the null hypothesis.

p-value : formally, the probability, under the null hypothesis of obtaining the observed value or any other value wich brings more evidence to reject ${H}_{0}$.

As an example let's say you want to test if a coin toss is actually fair or heads has a larger probability. In this case you would have:

${H}_{0}$: $p = {p}_{0} = 0.5$ vs ${H}_{1}$: $p > 0.5$, where p is the probability of the "heads" results.

Then you calculate your $\setminus \hat{p}$ wich in this case will be the proportion of heads in your sample. Let's say you got a 0.65 proportion in your sample.

Then your p-value will be the probability under ${H}_{0}$ (this means you calculate this probability assuming it's a fair coin) of having a proportion of 0.65 or more in your sample, because the larger the proportion, we get more evidence that "heads" has more probability.

Small p-values leads us to reject ${H}_{0}$.

## Questions

• · 2 months ago
• · 3 months ago
• 11 months ago
• 11 months ago
• · 2 years ago
• · 3 years ago