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

Testing claims about population means can be effectively done using confidence intervals or hypothesis tests. For a two-tailed test, if the claimed mean value is outside the $\mathrm{95\%}\mathrm{confidence\; interval}$, the null hypothesis is rejected. The critical value $z=1.96$ is used with margin of error $E=z\_\{\backslash alpha/2\}\; \backslash times\; \backslash frac\{\backslash sigma\}\{\backslash sqrt\{n\}\}$. Both methods yield consistent conclusions about the population mean. Confidence intervals are reliable for means and standard deviations but require caution for population proportions. Understanding these concepts aids in interpreting statistical evidence and making data-driven decisions.

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 \(α = 1 - 0.95 = 0.05\). The critical value, denoted as \(z_{\frac{α}{2}}\), corresponds to the z-score that captures the middle 95% of the standard normal distribution, which is approximately 1.96. The confidence interval is then constructed using the formula:

\[\bar{x} \pm z_{\frac{α}{2}} \times \frac{σ}{\sqrt{n}}\]

where \(\bar{x}\) is the sample mean, \(σ\) is the population standard deviation, and \(n\) is the sample size. The term \(z_{\frac{α}{2}} \times \frac{σ}{\sqrt{n}}\) represents the margin of error, which quantifies the range of uncertainty around the sample mean.

In a practical scenario, if the sample mean \(\bar{x}\) is 10, \(σ\) is 2, and \(n\) is 36, the margin of error is calculated as:

\[1.96 \times \frac{2}{\sqrt{36}} = 1.96 \times \frac{2}{6} = 1.96 \times 0.3333 \approx 0.65\]

This results in a confidence interval from \$10 - 0.65 = 9.35\( to \)10 + 0.65 = 10.65\(. If the claimed population mean, say 11, lies outside this interval, it suggests rejecting the null hypothesis that \)μ = 11\(.

Performing a two-tailed hypothesis test with the null hypothesis \)H_0: μ = 11\( and alternative hypothesis \)H_a: μ \neq 11\( involves calculating the test statistic using the z-score formula:

\[z = \frac{\bar{x} - μ_0}{\frac{σ}{\sqrt{n}}}\]

Substituting the values gives:

\[z = \frac{10 - 11}{\frac{2}{\sqrt{36}}} = \frac{-1}{\frac{2}{6}} = \frac{-1}{0.3333} = -3\]

The critical values for a 5% significance level in a two-tailed test are \)-1.96\( and \)1.96\(. Since the calculated z-score of -3 falls in the rejection region (less than -1.96), the null hypothesis is rejected, confirming the conclusion drawn from the confidence interval.

For one-tailed tests, the interpretation of confidence intervals adjusts accordingly. In a left-tailed test, the null hypothesis is rejected if the claimed value is entirely above the confidence interval, while in 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\() when the population standard deviation (\)σ\() is known, caution is needed when testing claims about population proportions (\)p$). Confidence intervals for proportions may not always provide the same clarity or reliability in hypothesis testing contexts.

Understanding the interplay between confidence intervals and hypothesis testing enhances the ability to make informed decisions about population parameters, reinforcing the foundational concepts of inferential statistics.

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. $\left\lbrack75.250,80.750\right\rbrack$
• B. $\left\lbrack75.948,80.052\right\rbrack$
• C. $\left\lbrack74.950,81.050\right\rbrack$
• D. $\left\lbrack74.550,81.450\right\rbrack$

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 $\mu\ne75$
• 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 population means?

Confidence intervals and hypothesis tests are closely related methods for testing claims about population means. When you construct a confidence interval, such as a 95% confidence interval, it provides a range of plausible values for the population mean. 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 (determined by the same confidence level), you reject the null hypothesis. Essentially, if the claimed mean is not within the confidence interval, the hypothesis test will also reject the claim, showing both methods lead to the same conclusion about the population mean.

What is the role of the critical value in linking 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 typically $z=1.96$. This value determines the margin of error in the confidence interval formula $E=z\cdot \sigma /\sqrt{n}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. In hypothesis testing, the same critical value defines the rejection region boundaries. If the test statistic exceeds these boundaries, the null hypothesis is rejected. Thus, the critical value links the two methods by setting consistent thresholds for decision-making about the population mean.

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

Confidence intervals are very effective for testing claims about population means and standard deviations, but caution is needed when applying them to population proportions. Unlike means, proportions have different distribution properties, especially with small sample sizes or proportions near 0 or 1. Building a confidence interval for a proportion may not always align perfectly with hypothesis testing results. Therefore, while confidence intervals can be used for proportions, it is important to verify assumptions and consider alternative methods like exact tests or adjusted intervals to ensure accurate conclusions.

How do you interpret a two-tailed hypothesis test using a confidence interval?

In a two-tailed hypothesis test, you test whether the population mean is equal to a claimed value versus it being different. When using a confidence interval, you construct, for example, a 95% interval around the sample mean. If the claimed mean lies within this interval, you do not reject the null hypothesis, indicating insufficient evidence to say the mean differs. However, if the claimed mean lies outside the interval, you reject the null hypothesis, concluding the mean is likely different. This interpretation matches the hypothesis test where the test statistic falls inside or outside the rejection region defined by the critical values.

What is the margin of error formula used in confidence intervals for population means?

The margin of error (E) in a confidence interval for a population mean is calculated using the formula $E=z\cdot \sigma /\sqrt{n}$. Here, $z$ is the critical value corresponding to the desired confidence level (e.g., 1.96 for 95%), $\sigma$ is the population standard deviation, and $n$ is the sample size. This margin of error determines how far the confidence interval extends from the sample mean, reflecting the uncertainty in estimating the population mean.