Cnx Is Retiring Textbooks By OpenStax Will Always Be Availab

Cnx Is Retiring Textbooks By Openstax Will Always Be Available At Op

Cnx is retiring! Textbooks by OpenStax will always be available at openstax.org. Community-created content will remain viewable until January 2022, and then be moved to Internet Archive. Conduct a hypothesis test using a preset α=0.05 to determine whether Jeffrey's new goggles helped him swim faster than his original mean time of 16.43 seconds, based on a sample mean of 16 seconds from 15 swims, assuming swim times are normal. Set up the hypotheses, determine the distribution, calculate the p-value, make a decision based on the significance level, and interpret the results. Repeat similar steps for other examples involving hypothesis testing for mean, proportion, and comparing sample data to hypothesized values with various significance levels.

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Hypothesis testing is a fundamental statistical procedure used to determine whether there is enough evidence in a sample to infer a particular characteristic about a population. In the context of Jeffrey’s swimming times, this involves testing whether his new goggles significantly improved his performance. The process begins with formulating the null hypothesis (H₀) and alternative hypothesis (H₁). Here, H₀ states that Jeffrey’s mean swim time remains at 16.43 seconds, while H₁ suggests it is less than that, indicating faster swimming due to the goggles.

Given the sample data—mean of 16 seconds, standard deviation of 0.8, and a sample size of 15—the appropriate distribution for testing the population mean is the normal distribution, especially since swim times are assumed to be normally distributed and the population standard deviation is known. The test statistic Z is computed as the difference between the sample mean and hypothesized mean, divided by the standard error. Using the sample data, Z = (16 - 16.43) / (0.8/√15) ≈ -2.08.

The p-value associated with this Z-score indicates the probability of observing a sample mean as extreme as or more extreme than 16 seconds, assuming H₀ is true. Consulting the standard normal distribution, a Z of -2.08 corresponds to a p-value of approximately 0.0187 for a one-tailed test. This p-value is less than the preset α of 0.05, leading to rejection of H₀ at the 5% significance level.

Interpreting these results, we conclude that there is statistically significant evidence that Jeffrey swims faster with his new goggles—his mean swim time is less than 16.43 seconds. This implies the goggles potentially have a beneficial effect on his performance, and the decrease in mean swim time is unlikely due to random variation alone.

The process for hypothesis testing is similar across different contexts. For example, in tests involving proportions, such as verifying if 50% of first-time brides are younger than their grooms, the sample proportion and the normal approximation for proportions are used. In one such case, with 53 out of 100 brides younger than grooms, the computed p-value was approximately 0.5485, which exceeds the significance level of 0.01. Therefore, we fail to reject the null hypothesis that the proportion is 50%.

Similarly, tests of population means when the population standard deviation is unknown require t-distributions, as illustrated by the exampl of students’ test scores. When sample sizes are large (n ≥ 30), the normal distribution approximation can be used; for smaller samples, the Student’s t-distribution applies, considering degrees of freedom (n - 1). For a sample mean of 67 with a standard deviation of about 3.2 from 10 students, the test statistic is approximately 1.978, and the p-value is approximately 0.0396. Since this p-value is less than 0.05, we reject H₀, concluding the average score is higher than 65 at the 5% significance level.

In all these examples, a critical element is the comparison between the p-value and the significance level α. If the p-value is less than α, we reject H₀, inferring that the evidence is sufficient to support the alternative hypothesis. Conversely, if the p-value exceeds α, we fail to reject H₀, indicating insufficient evidence to draw a conclusion.

Errors in hypothesis testing are also crucial to understanding the decision-making process. A Type I error occurs when we incorrectly reject the null hypothesis when it is true, potentially leading to false conclusions about the effect or difference. A Type II error occurs when we fail to reject the null hypothesis when the alternative is true, possibly missing genuine effects.

In conclusion, hypothesis testing offers a rigorous method for making inferences about populations based on sample data. Whether assessing the efficacy of new goggles, comparing means, or analyzing proportions, understanding the setup, calculation of the test statistic, p-value, and the decision rules are essential. Properly interpreting the results within the context of the significance level ensures meaningful and reliable conclusions in statistical analysis.

References

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