We Are Studying Hypothesis Testing My Instructor Asked Once
We Are Studying Hypothesis Testing My Instructor Asked Once The p Va
We are studying hypothesis testing. My instructor asked: Once the p-value is determined, how do you decide whether or not your evidence is strong enough? In other words, what is your criteria for rejecting the null hypothesis? Using the given confidence interval to find the estimated margin of error. Then find the sample mean. Commute times A government agency reports a confidence interval of (26.2, 30.1) when estimating the mean commute time (in minutes) for the population of workers in a city.
Paper For Above instruction
Hypothesis testing is a fundamental aspect of statistical inference, allowing researchers to determine whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. After calculating the p-value in a hypothesis test, researchers must decide whether the evidence is sufficiently strong to reject the null hypothesis. The p-value represents the probability of obtaining results as extreme as, or more extreme than, those observed, assuming that the null hypothesis is true. A common decision rule in hypothesis testing is to compare the p-value to a predetermined significance level, denoted as alpha (α), typically set at 0.05.
When the p-value is less than α, it indicates that the observed data is unlikely under the null hypothesis, and thus, the null hypothesis should be rejected. Conversely, if the p-value exceeds α, there is insufficient evidence to reject the null hypothesis, and it is retained. This comparison provides an objective criterion for decision-making in hypothesis testing. Moreover, the significance level (α) is chosen prior to conducting the test and reflects the researcher’s tolerance for Type I error—the probability of incorrectly rejecting a true null hypothesis. Therefore, the decision rule hinges on the p-value comparison: reject H₀ if p-value
In addition to p-values, confidence intervals offer valuable insight into the estimation of population parameters such as the mean. Given the confidence interval (26.2, 30.1) for the mean commute time, we can derive the estimated margin of error and the sample mean. The confidence interval is constructed as:
\[ \text{Sample Mean} \pm \text{Margin of Error} \]
From the interval bounds, we find:
\[
\text{Sample Mean} = \frac{26.2 + 30.1}{2} = 28.15
\]
The margin of error (ME) is half the width of the confidence interval:
\[
\text{ME} = \frac{30.1 - 26.2}{2} = 1.95
\]
This margin of error reflects the amount by which the sample mean estimate might differ from the true population mean with a specified confidence level.
Understanding the relationship between confidence intervals and hypothesis testing is crucial. For example, if a hypothesized population mean falls outside the 95% confidence interval, it suggests that this hypothesized value is unlikely given the data, leading to rejection of the null hypothesis at the corresponding significance level. Conversely, if the hypothesized value is within the interval, there is insufficient evidence to reject the null hypothesis.
In the context of commute times, the reported confidence interval suggests that, with 95% confidence, the true mean commute time for city workers lies between approximately 26.2 and 30.1 minutes. If a hypothesis test was conducted to evaluate whether the mean commute time equals a specific value—for instance, 28 minutes—comparison of this hypothesized value with the confidence interval guides the decision. Since 28 minutes falls within the interval, the evidence would not be sufficient to reject the null hypothesis that the true mean is 28 minutes.
In conclusion, the critical criterion for rejecting the null hypothesis based on p-values involves comparing the p-value to the significance level, generally rejecting H₀ if p-value
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