A one-proportion z-test is a statistical test used to compare the proportion of a sample to a known population proportion. It is used to test a hypothesis about the population proportion and is based on the assumption that the sample is drawn from a population with a normal distribution.

## Steps in One Proportion Z Test

To conduct a one-proportion z-test, the following steps are typically followed:

**Specify the null and alternative hypotheses.**The null hypothesis is usually that the population proportion is equal to a specific value, while the alternative hypothesis is that the population proportion is not equal to that value.**Select a sample**from the population and**calculate the sample proportion.****Calculate the test statistic,**which is the difference between the sample proportion and the population proportion, divided by the standard error of the proportion.- Determine the test statistic's
**critical value**based on the test's significance level (alpha). **Compare the calculated test statistic to the critical value**to determine whether to reject or fail to reject the null hypothesis. If the calculated test statistic exceeds the critical value, the null hypothesis is rejected, and the alternative hypothesis is accepted.

## Conditions for Z-Test

To conduct a valid one-proportion z-test, the following conditions must be met:

- The sample must be
**drawn randomly**from the population. - Each observation in the sample must be
**independent of the others.**- The sampling must be done with replacement, OR
- If sampling without replacement, the sample size must be less than 10% of the population.

- The data contains only two categories: (for example) pass/fail or yes/no.
- For Normal approximation:

Both np ≥ 10 and n(1−p) ≥ 10 (data should have at least 10 "successes" and at least 10 "failures" ) (in some books, it is 5)

## Typical Null and Alternate Hypothesis in One Proportion Z-Test

**a) Two-Tail Test:** In a one-proportion z-test, the null hypothesis is that there is no difference between the proportion of the sample and the known population proportion. This can be expressed as:

H0: p = p0

where p is the proportion of the sample and p0 is the known population proportion.

The alternate hypothesis is the opposite of the null hypothesis and is that there is a difference between the proportion of the sample and the known population proportion. This can be expressed as:

Ha: p ≠ p0

**b) Left Tail Test:** A left-tailed hypothesis is one in which the sample proportion is less than the known population proportion. This can be expressed as:

H0: p >= p0

Ha: p < p0

**c) Right Tail Test:** A right-tailed hypothesis is one in which the sample proportion is greater than the known population proportion. This can be expressed as:

H0: p <= p0

Ha: p > p0

## Calculating Test Statistic

The z-score represents the number of standard errors that the sample proportion is from the population proportion. It is used to determine whether the difference between the sample and population proportions is statistically significant. The formula for calculating the z-score in a one-proportion z-test is as follows:

$$\LARGE{z = \frac{(p - p0)}{\sqrt{\frac{p0(1 - p0)}{n}}}}$$

Where p is the sample proportion, p0 is the population proportion, and n is the sample size.

## Calculating Critical Values

The critical values for the z-score in a one-proportion z-test depend on whether the test is a one-tail or two-tail test.

For a one-tail test, the critical value is determined based on the tail of the distribution in which the alternative hypothesis is located. For example, if the alternative hypothesis is that the population proportion is greater than a specific value, the critical value would be the value that corresponds to the upper tail of the distribution.

For a two-tail test, the critical value is determined based on the significance level of the test and the total area of both tails of the distribution. For example, if the significance level is 0.05 and the test is two-tailed, the critical value would be the value that corresponds to an area of 0.05 in both tails of the distribution (or 0.025 area in one tail, as the normal distribution is symmetric).

The most common critical values for the z-score in a one-sample z-test are as follows:

- For a one-tail test with a significance level of 0.05, the critical value is 1.64.
- For a one-tail test with a significance level of 0.01, the critical value is 2.33.
- For a one-tail test with a significance level of 0.001, the critical value is 3.09.
- For a two-tail test with a significance level of 0.05, the critical value is 1.96.
- For a two-tail test with a significance level of 0.01, the critical value is 2.58.
- For a two-tail test with a significance level of 0.001, the critical value is 3.29.

These critical values are based on the standard normal distribution and are used to determine whether the calculated z-score is statistically significant. If the calculated z-score is greater than the critical value, the null hypothesis is rejected, and the alternative hypothesis is accepted.

You can find the critical values using a Normal Distribution table or software such as Excel. The formula to find out this value in Excel is **=NORM.S.INV(alpha)** for the one-tail test and **=NORM.S.INV(alpha/2)** for the two-tail test. This provides the critical z value for the left tail.