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. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters, such as \(p, \mu_{1}-\mu_{2}\), and \(p_{1}-p_{2}\), which we will study in Sections \(8.3\) and \(8.5 .\).) 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 7, 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 this chapter, 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

The relationship between confidence intervals and two-tailed hypothesis tests is that the confidence level, denoted as \(c\), is determined by the equation \(c = 1 - \alpha\). Here, \(\alpha\) is given as 0.01, so the confidence level \(c\) is calculated as \(c = 1 - 0.01 = 0.99\).

02

Construct Confidence Interval

To construct a \(99\%\) confidence interval for the mean \(\mu\), we use the formula: \[\bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right)\] where \(\bar{x} = 22\), \(\sigma = 4\), \(n = 36\), and \(z\) is the critical value for \(99\%\) confidence. The critical \(z\)-value for \(0.005\) in each tail (since it's two-tailed) is approximately \(2.576\). Therefore, the margin of error is \(2.576 \times \frac{4}{\sqrt{36}} = 2.576 \times \frac{4}{6} = 1.717\). The confidence interval is \([22 - 1.717, 22 + 1.717] = [20.283, 23.717]\).

03

Compare k with Confidence Interval

In the null hypothesis, \(k = 20\). We need to check if this value is within the confidence interval \([20.283, 23.717]\). Since \(20\) is outside this interval, it suggests we reject the null hypothesis \(H_0: \mu = 20\).

04

Calculate P-value

To find the \(P\)-value, we compute the test statistic using \(z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\) where \(\mu_0 = 20\). The test statistic is \(z = \frac{22 - 20}{4/\sqrt{36}} = \frac{2}{\frac{2}{3}} = 3\). Looking up \(z = 3\) in a standard normal table gives a one-tailed area of approximately \(0.0013\), and thus the \(P\)-value for both tails is \(2 \times 0.0013 = 0.0026\).

05

Decision from P-value

The \(P\)-value \(0.0026\) is less than \(\alpha = 0.01\). This indicates strong evidence against the null hypothesis, so we reject \(H_0\). This matches the conclusion from Step 3 where the null hypothesis was outside the confidence interval.

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Key Concepts

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

Two-Tailed Hypothesis Tests

Two-tailed hypothesis tests examine whether a sample mean differs significantly from a hypothesized mean, considering both directions of difference. In a two-tailed test, the null hypothesis asserts that there is no effect or difference, while the alternative hypothesis suggests that there is a significant effect either larger or smaller than the hypothesized mean. With this format, deviations on both ends of the distribution are of interest. For instance, if \( H_0: \mu = 20 \) this means we're testing if the mean could significantly be greater or less than 20. The test considers both possibilities by evaluating if the test statistic falls in either tail of the distribution. This is crucial when you're interested in any change, not simply whether the mean increases or decreases. Your goal in a two-tailed test is to determine if the sample mean falls significantly outside a specified range around the null mean.

Level of Significance

The level of significance, denoted as \( \alpha \), is a threshold we set for deciding whether to reject the null hypothesis. This value reflects the probability of making a Type I error—rejecting a true null hypothesis. Commonly used significance levels are 0.05, 0.01, and 0.10, among others. For example, in our exercise, \( \alpha = 0.01 \), is quite stringent, indicating that the researcher only tolerates a 1% chance of accepting a false positive. Lower \( \alpha \) means stricter criteria for rejecting the null hypothesis, which is useful when the cost of such an error is high. Based on the selected \( \alpha \), the critical values are determined, defining the boundaries beyond which the null hypothesis is rejected.

P-Value

The P-value is a statistical metric indicating the probability of obtaining an observation at least as extreme as the current data, given that the null hypothesis is true. Lower P-values suggest stronger evidence against the null hypothesis. In our example, we compute the P-value using the test statistic and the standard normal distribution. A key point here is the comparison between the P-value and the level of significance \( \alpha \). If the P-value is less than \( \alpha \), then the result is considered statistically significant, and \( H_0 \) is rejected. In our step-by-step solution, we found a P-value of 0.0026, which is significantly lower than \( \alpha = 0.01 \), hence supporting the rejection of the null hypothesis, aligning with our confidence interval findings.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of zero and a standard deviation of one. It is represented by the \( z \)-score scale, which facilitates the determination of critical values and P-values. In hypothesis testing, especially in a two-tailed test, \( z \)-scores help identify how many standard deviations an observation is from the mean. This is especially useful in drawing conclusions from statistical data. In our scenario, the standard normal distribution allows us to derive the critical \( z \)-value of approximately 2.576 for a 99% confidence interval, vital in making decisions about the null hypothesis. Converting sample means into \( z \)-scores provides a standardized method to make data-driven decisions.