Link Between Confidence Intervals and Hypothesis Testing: Videos & Practice Problems

Testing claims about a population mean can be done using confidence intervals or hypothesis tests. A 95% confidence interval is constructed using the sample mean ($x¯$), population standard deviation (σ), sample size (n), and critical value (z_α/2). If the claimed mean is outside this interval, the null hypothesis is rejected. Hypothesis testing uses the test statistic $\frac{x¯-\mu }{\sigma /\sqrt{n}}$ compared to critical z-values. Both methods provide consistent conclusions about population means.

Link Between Confidence Intervals and Hypothesis Testing

Testing claims about population means can be effectively done using either hypothesis testing or confidence intervals, as both methods lead to the same conclusion when applied correctly. When constructing a confidence interval for the population mean, the key is to check whether the claimed value lies within this interval. If the claimed value is outside the confidence interval, it indicates that the null hypothesis would be rejected in a corresponding hypothesis test, suggesting the claim is unlikely to be true.

For example, when creating a 95% confidence interval for the population mean μ, the significance level α is calculated as 1 − confidence level, which in this case is 0.05. The critical value z_α/2 corresponds to the z-score that captures the middle 95% of the standard normal distribution, approximately 1.96. The margin of error E is computed using the formula:

\[E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\]

where σ is the population standard deviation and n is the sample size. The confidence interval is then:

\[\bar{x} - E \quad \text{to} \quad \bar{x} + E\]

For instance, with a sample mean \bar{x} of 10, population standard deviation σ of 2, and sample size n of 36, the margin of error is approximately 0.65, resulting in a confidence interval from 9.35 to 10.65. Since the claimed value of 11 lies outside this interval, the claim that μ = 11 is rejected.

Performing a two-tailed hypothesis test for the same claim involves setting the null hypothesis H₀: μ = 11 and the alternative hypothesis Hₐ: μ ≠ 11. Using the critical value method with α = 0.05, the rejection regions are defined as values less than −1.96 or greater than 1.96 for the test statistic:

\[z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]

Substituting the values gives:

\[z = \frac{10 - 11}{2 / \sqrt{36}} = \frac{-1}{2 / 6} = \frac{-1}{\frac{1}{3}} = -3\]

Since −3 falls within the rejection region, the null hypothesis is rejected, confirming the conclusion drawn from the confidence interval.

When dealing with one-tailed tests, the interpretation of confidence intervals adjusts accordingly. For a left-tailed test, the null hypothesis is rejected if the claimed value is entirely above the confidence interval, while for a right-tailed test, rejection occurs if the claimed value is entirely below the confidence interval.

It is important to note that while confidence intervals and hypothesis tests align well for population means (μ) and population standard deviations (σ), caution is needed when testing claims about population proportions (p). Confidence intervals for proportions may not always provide the intended results due to differences in distribution and sample size considerations.

Understanding the relationship between confidence intervals and hypothesis testing enhances the ability to analyze statistical claims effectively, providing a robust framework for decision-making based on sample data.

A teacher claims her students’ average test score is 75. A researcher suspects it’s different. A sample of 25 students has a mean score of 78 with a standard deviation of 6.
$\left(A\right)$ Create a $90\%$ confidence interval for the mean test score.

• A. $[75.250,80.750]$
• B. $[75.948,80.052]$
• C. $[74.950,81.050]$
• D. $[74.550,81.450]$

Answer: (B)

A teacher claims her students’ average test score is 75. A researcher suspects it’s different. A sample of 25 students has a mean score of 78 with a standard deviation of 6.
$\left(B\right)$ At the $0.10$ significance level, test the claim.

• A. $P-value=0.02$, fail to reject $H_0$. There is not enough evidence to suggest $\mu\ne75$
• B. $P-value=0.08$, reject $H_0$. There is enough evidence to suggest $\mu\ne75$
• C. $P-value=0.02$, reject $H_0$. There is enough evidence to suggest $\mu \ne 75$
• D. $P-value=0.08$, fail to reject $H_0$. There is not enough evidence to suggest $\mu\ne75$

Answer: (C)

Here’s what students ask on this topic:

How are confidence intervals and hypothesis tests related when testing a population mean?

Confidence intervals and hypothesis tests are closely related methods for testing claims about a population mean. When you build a confidence interval, such as a 95% confidence interval, you calculate a range of plausible values for the population mean based on your sample data. If the claimed mean value lies outside this interval, it suggests that the claim is unlikely to be true. This corresponds directly to hypothesis testing, where if the test statistic falls in the rejection region, you reject the null hypothesis. Essentially, if the claimed mean is not within the confidence interval, the hypothesis test would also reject the null hypothesis, leading to the same conclusion about the population mean.

What is the role of the critical value in both confidence intervals and hypothesis testing?

The critical value plays a key role in both confidence intervals and hypothesis testing. For a 95% confidence interval, the critical value is $z{\alpha }_2=1.96$, which determines the margin of error around the sample mean. This margin of error defines the range of the confidence interval. In hypothesis testing, the same critical value defines the rejection regions on both tails of the distribution for a two-tailed test. If the test statistic falls beyond these critical values, the null hypothesis is rejected. Thus, the critical value links the confidence interval's range and the hypothesis test's decision boundaries.

Can confidence intervals be used to test claims about population proportions as effectively as for population means?

Confidence intervals work well for testing claims about population means and standard deviations, but they are less straightforward for population proportions. While you can build confidence intervals for proportions, the interpretation and accuracy depend on sample size and distribution assumptions. Sometimes, the confidence interval method may not align perfectly with hypothesis testing results for proportions, especially with small samples or extreme proportions. Therefore, caution is needed when using confidence intervals to test claims about population proportions, and hypothesis testing methods specifically designed for proportions might be more reliable.

How do left-tailed and right-tailed hypothesis tests relate to confidence intervals?

Left-tailed and right-tailed hypothesis tests relate to confidence intervals by the position of the claimed value relative to the interval. In a left-tailed test, you reject the null hypothesis if the claimed value is entirely above the confidence interval, indicating the population mean is likely less than the claim. Conversely, in a right-tailed test, you reject the null hypothesis if the claimed value is entirely below the confidence interval, suggesting the population mean is likely greater than the claim. This relationship helps use confidence intervals to make decisions in one-tailed hypothesis tests.

What steps are involved in using a confidence interval to test a claim about a population mean?

To use a confidence interval to test a claim about a population mean, follow these steps: First, determine the confidence level (e.g., 95%) and calculate the corresponding alpha (α = 1 - confidence level). Next, find the critical value $z{\alpha }_2$ using a z-table or technology. Then, calculate the margin of error using the formula $E=z{\alpha }_2\times \sigma /\sqrt{n}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Construct the confidence interval by adding and subtracting the margin of error from the sample mean. Finally, check if the claimed mean lies within this interval. If it does not, reject the claim; if it does, do not reject it.