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A confidence interval is a range of values used to estimate a population parameter. In the context of hypothesis testing, it provides an estimated range where we expect the true population mean to fall. The calculation of a confidence interval involves the sample mean, the standard deviation, and the critical t-value related to the selected confidence level.
For instance, in our example, a 95% confidence interval was computed using the following formula: \( \bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right) \).
This includes:

• \( \bar{x} = 98.285 \), the sample mean temperature.
• \( s = 0.625 \), the standard deviation of the sample.
• \( n = 52 \), the sample size.
• \( t_{\alpha/2} = 2.008 \), the critical value from the t-distribution table at \( \alpha = 0.05 \).

This process resulted in a confidence interval of approximately (98.112, 98.458). Since the supposedly true mean of 98.6°F is outside this range, it provides evidence against the null hypothesis, suggesting that 98.6°F is not a reasonable estimate of the population mean.

The t-statistic is a ratio that comes into play when you want to decide whether to reject a null hypothesis in hypothesis testing. It compares the observed sample mean to the population mean under the null hypothesis, scaling by the variability observed in the sample and the size of the sample.
The t-statistic formula is: \[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]where:

• \( \bar{x} \) is the sample mean
• \( \mu \) is the population mean under the null hypothesis
• \( s \) is the standard deviation of the sample
• \( n \) is the sample size

In our example, substituting the values gives us a t-statistic of approximately \(-3.629\). This t-value is then compared to a critical value from the t-distribution with \( n-1 \) degrees of freedom.
The t-statistic tells us how far our sample mean is from the hypothesized population mean, in units of standard error. A high absolute value of the t-statistic indicates a large deviation from the null hypothesis.

In hypothesis testing, we work with two main hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis (\( H_0 \)) represents the default or original assumption, which, in many cases, claims that there is no effect or difference.
In this exercise, the null hypothesis is that the mean oral body temperature is 98.6°F, or \( H_0: \mu = 98.6 \).
The alternative hypothesis (\( H_a \)) reflects the researcher's question or suspicion that contradicts the null hypothesis. It posits that there is an effect or a difference.
For this example, the alternative hypothesis is that the mean is not 98.6°F, written as \( H_a: \mu eq 98.6 \).
These hypotheses help guide the entire statistical test, determining whether the sample data provides sufficient evidence to reject the null hypothesis.

A two-tailed test is a type of hypothesis test where we examine the possibility of the sample mean being either significantly higher or lower than the hypothesized population mean.
This is in contrast to a one-tailed test where we test the direction of the deviation.
In a two-tailed test, the null hypothesis is tested against an alternative that allows for deviations on both sides.
For the task at hand, the hypotheses \( H_0: \mu = 98.6 \) and \( H_a: \mu eq 98.6 \) mean that we are interested in whether the actual mean temperature significantly deviates from 98.6°F, be it higher or lower.

• Critical regions are located on both extremes of the distribution.
• This approach increases the range of conditions under which the null hypothesis can be rejected.

In this way, a two-tailed test provides a more comprehensive assessment for scenarios where deviations may occur in either direction.