91Ó°ÊÓ

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let \(c\) be the level of confidence used to construct a confidence interval from sample data. Let \(\alpha\) be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. For a two-tailed hypothesis test with level of significance \(\alpha\) and null hypothesis \(H_{0}: \mu=k\), we reject \(H_{0}\) whenever \(k\) falls outside the \(c=1-\alpha\) confidence interval for \(\mu\) based on the sample data. When \(k\) falls within the \(c=1-\alpha\) confidence interval, we do not reject \(H_{0}\). (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as \(p, \mu_{1}-\mu_{2}\), or \(p_{1}-p_{2}\) which we will study in Sections \(9.3\) and \(9.5 .\) ) Whenever the value of \(k\) given in the null hypothesis falls outside the \(c=1-\alpha\) confidence interval for the parameter, we reject \(H_{0} .\) For example, consider a two-tailed hypothesis test with \(\alpha=0.01\) and \(H_{0}: \mu=20 \quad H_{1}: \mu \neq 20\) A random sample of size 36 has a sample mean \(\bar{x}=22\) from a population with standard deviation \(\sigma=4\). (a) What is the value of \(c=1-\alpha\) ? Using the methods of Chapter 8 , construct a \(1-\alpha\) confidence interval for \(\mu\) from the sample data. What is the value of \(\mu\) given in the null hypothesis (i.e., what is \(k)\) ? Is this value in the confidence interval? Do we reject or fail to reject \(H_{0}\) based on this information? (b) Using methods of Chapter 9 , find the \(P\) -value for the hypothesis test. Do we reject or fail to reject \(H_{0}\) ? Compare your result to that of part (a).

Step by step solution

01

Calculate Confidence Level

Given \( \alpha = 0.01 \), the confidence level \( c = 1 - \alpha = 0.99 \). This indicates a 99% confidence interval for the population mean \( \mu \).

02

Construct 99% Confidence Interval for \( \mu \)

Since the population standard deviation is known (\( \sigma = 4 \)), we use the normal distribution. The sample size is 36, and the sample mean is 22. The z-score for a 99% confidence interval is approximately 2.576. The confidence interval is given by: \[ CI = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) = 22 \pm 2.576 \left( \frac{4}{6} \right), \] which simplifies to \( 22 \pm 1.717 \), resulting in \( (20.283, 23.717) \).

03

Evaluate Null Hypothesis using Confidence Interval

The null hypothesis states \( \mu = 20 \). This value (20) lies outside the confidence interval (20.283, 23.717), so we reject \( H_0 \).

04

Find P-value for Hypothesis Test

To find the P-value, we need the test statistic. The formula for the z-test statistic for \( \mu \) is \[ z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} = \frac{22 - 20}{\frac{4}{6}} = 3. \] For a two-tailed test, the P-value is \( 2 \times P(Z > 3) \). Looking up a z-table, \( P(Z > 3) \approx 0.0013 \), so the P-value is approximately \( 2 \times 0.0013 = 0.0026 \).

05

Decision based on P-value

Since the P-value (0.0026) is less than \( \alpha = 0.01 \), we reject \( H_0 \).

06

Compare Results from Confidence Interval and P-value

Both methods lead to rejecting \( H_0 \). The value \( \mu = 20 \) is outside the 99% confidence interval, and the P-value is less than the significance level. Thus, the decision to reject \( H_0 \) is consistent with both approaches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Confidence Intervals

Confidence intervals are a fundamental concept in statistics, representing a range of values that is likely to contain a population parameter with a certain level of confidence. In our exercise, this parameter is the population mean \( \mu \). The confidence interval provides a range that estimates the value of \( \mu \) based on sample data.
It's calculated using the formula: \[ CI = \bar{x} \pm z \left( \frac{\sigma}{\sqrt{n}} \right) \] where \( \bar{x} \) is the sample mean, \( \sigma \) is the population standard deviation, \( n \) the sample size, and \( z \) the z-score associated with the desired confidence level.
The confidence level (\( c \)), often expressed as a percentage, denotes the probability that the interval encompasses the true parameter. For instance, a 99% confidence interval suggests a 99% chance that the interval includes the true mean. The higher the confidence level, the wider the interval becomes since it must account for more uncertainty.

Two-tailed Hypothesis Test

A two-tailed hypothesis test is used to determine if a sample mean significantly differs from a hypothesized population mean. This test checks for deviations in both directions—whether the sample mean is significantly greater or less than the hypothesized mean.
In our example, the null hypothesis (\( H_0: \mu = 20 \)) posits that the population mean is 20. The alternative hypothesis (\( H_1: \mu eq 20 \)), on the other hand, is concerned if the mean is not equal to 20, thus allowing for testing in two 'tails' of the distribution.
This type of test is particularly useful in scenarios where we are interested in any substantial difference, regardless of direction, between a sample statistic and a population parameter. A critical aspect is the level of significance \( \alpha \), which is divided equally across both tails of the distribution for making decisions.

Level of Significance

The level of significance, \( \alpha \), is a threshold set in hypothesis tests to decide whether to reject the null hypothesis. It's the probability of rejecting \( H_0 \) when it is actually true, essentially describing the risk of a Type I error.
In hypothesis testing, common significance levels are 0.05, 0.01, or 0.10, which translate to 5%, 1%, or 10% risk levels, respectively. The choice of \( \alpha \) reflects how much uncertainty a researcher is willing to accept. A smaller \( \alpha \) implies stricter criteria for rejecting \( H_0 \), offering more confidence in the results at the expense of potentially missing a true effect.
Our original exercise utilized \( \alpha = 0.01 \), indicating a stringent standard, where the null hypothesis is only rejected if results are extremely unlikely assuming \( H_0 \) is true. This level of significance is therefore crucial in defining the boundaries of critical regions in hypothesis testing.

P-value

The P-value is an essential component in hypothesis testing, offering a way to measure the strength of the evidence against the null hypothesis. It represents the probability of observing a test statistic at least as extreme as the one calculated, given that the null hypothesis is true.
A smaller P-value indicates stronger evidence against \( H_0 \), often prompting the rejection of the null hypothesis. In our example, the calculated P-value was 0.0026, which was less than the significance level \( \alpha = 0.01 \), guiding us to reject \( H_0 \).
Typically, if the P-value is below \( \alpha \), it suggests the observed data are unlikely under \( H_0 \), supporting the need to consider the alternative hypothesis. Thus, the P-value functions as a crucial decision-making tool, pinpointing how far the sample evidence contradicts the null hypothesis.