One-Sample Z-Test for a Proportion

Definition

The one-sample z-test for a proportion is a statistical hypothesis test used to determine if the proportion of a characteristic in a population is equal to a hypothesized or known value. It is commonly used when the sample size is large and the population standard deviation is known.

5 Must Know Facts For Your Next Test

1. The one-sample z-test for a proportion is used when the population standard deviation is known and the sample size is large (typically greater than 30).
2. The test statistic for the one-sample z-test for a proportion is calculated as $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size.
3. The null hypothesis for the one-sample z-test for a proportion is that the population proportion is equal to the hypothesized value ($H_0: p = p_0$), while the alternative hypothesis can be either one-sided ($H_a: p > p_0$ or $H_a: p < p_0$) or two-sided ($H_a: p \neq p_0$).
4. The test statistic follows a standard normal distribution (z-distribution) under the null hypothesis, allowing for the use of z-tables or z-calculators to determine the p-value and make a decision about the hypothesis.
5. The one-sample z-test for a proportion is used to make inferences about a population proportion when the sample size is large enough to assume a normal sampling distribution, even if the underlying population distribution is not normal.

Review Questions

• Explain the purpose and assumptions of the one-sample z-test for a proportion.
  • The one-sample z-test for a proportion is used to determine if the proportion of a characteristic in a population is equal to a hypothesized or known value. The key assumptions are that the sample size is large (typically greater than 30) and the population standard deviation is known. Under these conditions, the test statistic follows a standard normal distribution, allowing for the use of z-tables or z-calculators to make inferences about the population proportion.
• Describe the test statistic and hypotheses used in the one-sample z-test for a proportion.
  • The test statistic for the one-sample z-test for a proportion is calculated as $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and $n$ is the sample size. The null hypothesis is that the population proportion is equal to the hypothesized value ($H_0: p = p_0$), while the alternative hypothesis can be either one-sided ($H_a: p > p_0$ or $H_a: p < p_0$) or two-sided ($H_a: p \neq p_0$).
• Explain how the results of the one-sample z-test for a proportion are interpreted and used to make decisions about the population proportion.
  • The results of the one-sample z-test for a proportion are used to determine the likelihood of observing the sample proportion if the null hypothesis is true. The p-value, which represents the probability of obtaining a test statistic at least as extreme as the one observed, is compared to the chosen significance level (typically 0.05 or 0.01). If the p-value is less than the significance level, the null hypothesis is rejected, and it is concluded that the population proportion is not equal to the hypothesized value. The direction of the alternative hypothesis (one-sided or two-sided) determines the specific conclusion about the population proportion.

Related terms

Hypothesis Testing: A statistical method used to determine whether a claim or hypothesis about a parameter of a population is likely to be true.

Proportion: The ratio of the number of successes to the total number of trials in a sample or population.

Z-score: A standardized measure of how many standard deviations a data point is from the mean of a normal distribution.