Sample Proportion

Definition

The sample proportion is the proportion or percentage of a sample that exhibits a particular characteristic or attribute of interest. It is a crucial concept in statistics that is used to make inferences about the true proportion or percentage of a population based on a representative sample.

5 Must Know Facts For Your Next Test

1. The sample proportion is denoted by the symbol $\hat{p}$, where $\hat{}$ (hat) indicates that it is an estimate of the true population proportion.
2. The sample proportion is calculated by dividing the number of successes (individuals in the sample that exhibit the characteristic of interest) by the total sample size.
3. The Central Limit Theorem for Proportions states that the sampling distribution of the sample proportion will be approximately normal, with a mean equal to the true population proportion and a standard deviation that decreases as the sample size increases.
4. A confidence interval for a population proportion is used to estimate the true population proportion based on the sample proportion and the sample size.
5. Hypothesis testing for a population proportion involves comparing the sample proportion to a hypothesized population proportion to determine if there is sufficient evidence to reject the null hypothesis.

Review Questions

• Explain how the sample proportion is used in the context of the Central Limit Theorem for Sample Means.
  • The Central Limit Theorem for Sample Means states that as the sample size increases, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. This theorem also applies to the sampling distribution of the sample proportion, $\hat{p}$. When the sample size is sufficiently large, the sampling distribution of $\hat{p}$ will be approximately normal, with a mean equal to the true population proportion, $p$, and a standard deviation that decreases as the sample size increases. This allows for the use of the normal distribution to make inferences about the population proportion based on the sample proportion.
• Describe how the sample proportion is used to construct a confidence interval for a population proportion.
  • To construct a confidence interval for a population proportion, $p$, based on a sample proportion, $\hat{p}$, we use the formula: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $n$ is the sample size and $z_{\alpha/2}$ is the critical value from the standard normal distribution corresponding to the desired confidence level. This confidence interval provides a range of plausible values for the true population proportion, $p$, based on the sample proportion and the sample size. The Central Limit Theorem for Proportions ensures that this approach is valid when the sample size is sufficiently large.
• Explain how the sample proportion is used in the context of comparing two independent population proportions.
  • When comparing two independent population proportions, $p_1$ and $p_2$, the sample proportions, $\hat{p}_1$ and $\hat{p}_2$, are used to test the null hypothesis that the two population proportions are equal ($H_0: p_1 = p_2$) against the alternative hypothesis that they are not equal ($H_A: p_1 \neq p_2$). The test statistic used is the difference between the two sample proportions, $\hat{p}_1 - \hat{p}_2$, which follows an approximately normal distribution when the sample sizes are sufficiently large. This allows for the use of the standard normal distribution to calculate the p-value and make a decision about the null hypothesis, ultimately drawing conclusions about the relationship between the two population proportions.

Related terms

Population Proportion: The true proportion or percentage of a characteristic or attribute in the entire population being studied.

Sampling Distribution: The distribution of a statistic, such as the sample proportion, across all possible samples drawn from a population.

Central Limit Theorem: A fundamental theorem in statistics that states that the sampling distribution of the sample mean (or proportion) will be approximately normal, regardless of the shape of the population distribution, as the sample size increases.