One-sample z-interval

Definition

A one-sample z-interval is a statistical method used to estimate the confidence interval for a population proportion based on a single sample. It leverages the normal distribution to make inferences about the population, relying on the sample proportion, the sample size, and the desired level of confidence. This technique is particularly effective when the sample size is large enough to meet the conditions of normality and when the success-failure condition is satisfied, ensuring that both np and n(1-p) are greater than 5.

5 Must Know Facts For Your Next Test

1. To construct a one-sample z-interval for a population proportion, you use the formula: \(\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(z^*\) corresponds to the critical value for your chosen confidence level.
2. The success-failure condition states that both \(np\) and \(n(1-p)\) must be greater than 5 for the normal approximation to be valid.
3. A higher confidence level results in a wider interval because it accounts for more variability in the population proportion estimation.
4. The one-sample z-interval is only appropriate when the sample size is sufficiently large; if not, other methods like the exact binomial interval may be used.
5. Interpreting a one-sample z-interval involves stating that you are confident that the true population proportion lies within the calculated interval based on your sample data.

Review Questions

• What steps would you take to calculate a one-sample z-interval for a population proportion?
  • To calculate a one-sample z-interval, first collect your sample data and determine the sample proportion \(\hat{p}\). Next, check that your sample size meets the success-failure condition by ensuring both \(np\) and \(n(1-p)\) are greater than 5. Then choose your desired confidence level and find the corresponding critical value \(z^*\). Finally, use the formula \(\hat{p} \pm z^* \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) to compute the interval.
• How does changing the confidence level affect the width of a one-sample z-interval?
  • Changing the confidence level directly affects the width of a one-sample z-interval. A higher confidence level means you want to be more certain that your interval contains the true population proportion, which results in a larger critical value \(z^*\). Consequently, this leads to a wider interval. Conversely, lowering the confidence level reduces the width of the interval but also decreases your certainty about capturing the true parameter.
• Evaluate why it's important to check conditions before applying the one-sample z-interval method.
  • Checking conditions before applying the one-sample z-interval method is crucial because it ensures that our results are reliable and valid. If we skip this step and apply the method without meeting the success-failure condition or having a sufficiently large sample size, we risk producing misleading intervals that do not accurately reflect the population proportion. Validating these conditions preserves the integrity of our statistical inference and allows us to make sound decisions based on our analysis.

Related terms

Confidence Level:

The probability that the interval estimate will contain the true population parameter, typically expressed as a percentage, such as 90%, 95%, or 99%.

Sample Proportion:

The ratio of the number of successes in a sample to the total number of observations in that sample, often denoted as \(\hat{p}\).

Standard Error:

The estimated standard deviation of the sampling distribution of a statistic, which quantifies how much the sample statistic is expected to vary from the actual population parameter.