Unit 5: Assignment Hypothesis Testing and Errors

HS 4510 - 01 Biostatistics - UNIT 5 Written Assignment Instructor: Pankaj Mehrotra Date: 02/03/2025 QUESTION 1 Null Hypothesis (H₀): The soft drink has acceptable sugar levels, meaning the sugar content is within the acceptable limit. Alternative Hypothesis (H₁): The soft drink has higher sugar levels than what is acceptable. Type 1 Error: Rejecting the null hypothesis wrongly is a Type 1 error. A Type 1 error in this instance would be the researcher's conclusion that the soft drink has more sugar than is reasonable (rejecting H₀) when, in reality, the sugar levels are within the allowed range. This would result in an unwarranted worry or action based on flawed data. Type 2 Error: When the null hypothesis is accepted despite being incorrect, this is known as a Type 2 error. A Type 2 error in this instance would be the researcher's failure to recognize that the soft drink has more sugar than is reasonable (failing to reject H₀), even if it does. As a result, the problem would go unnoticed, and the elevated sugar levels would not be addressed. QUESTION 2 Changing the significance level from 0.05 to 0.01 has the following effects: Type 1 Error: A Type 1 error (false positive) is less likely to occur. Rejecting the null hypothesis at a lower significance level indicates greater conservatism. The researcher would need more proof to reject the null hypothesis at a significance level of 0.01 than at a 0.05 level. As a result, there is less chance of making a Type 1 error—erroneously rejecting a true null hypothesis. Type 2 Error: A Type 2 error (false negative) is more likely to occur. Even if the null hypothesis is incorrect, you are less likely to reject it because the significance level is lowered. As a result, the study becomes more cautious, and there is a greater probability of Type 2 error if the effect is not large enough for the test to identify a true difference. Power: The test's power declines. The ability to accurately reject a false null hypothesis is known as power, and higher evidence is needed to reject the null hypothesis as the significance threshold is lowered. A smaller alpha (0.01) lowers the test's overall power since it makes it less sensitive to identifying minute variations or effects. In summary: ●​ Type 1 error: Decreases ●​ Type 2 error: Increases ●​ Power: Decreases QUESTION 3 A number of statistical factors, such as the likelihood of Type 1 and Type 2 errors and the study's power, are impacted when the confidence interval (CI) is reduced from 95% to 90%. Type 1 Error: Effect: Increase The explanation is that a significance level of 0.10, or a 90% confidence interval, means that there is a 10% chance of rejecting the null hypothesis when it is true. On the other hand, a 95% CI is equivalent to a 0.05 significance threshold, or a 5% chance. You are more likely to reject the null hypothesis when it is true because the threshold for doing so is loosened when the confidence level is reduced to 90%. You are more likely to make a Type 1 error as a result of this since you are more likely to make an incorrect association claim. Type 2 Error: Effect: Decrease Justification: It is simpler to reject the null hypothesis when the confidence interval is smaller (90 percent CI). As a result, the test is better able to identify a real effect. This reduces the likelihood of a Type 2 error since you are less likely to fail to reject the null hypothesis when there is a true effect. Stated differently, you have a higher chance of correctly rejecting the null hypothesis when it is incorrect. Power of the Study: Effect: Increase Power is defined as the likelihood of correctly rejecting the null hypothesis in the event that it is incorrect. Lowering the confidence threshold to 90% boosts the study's power since it makes it more likely that the null hypothesis will be rejected, even if the effect is little. With a more relaxed significance criterion (alpha = 0.10), the study has a higher chance of identifying a real effect, if one exists. Summary of Effects: Type 1 error: Increases (more likely to reject the null hypothesis when it is true) Type 2 error: Decreases (less likely to miss a true effect) Power: Increases (greater likelihood of detecting a true effect) QUESTION 4 Research Project Design Research Question: According to changes in resting heart rate at baseline, two months, and four months following the intervention, does a four-month physical fitness program enhance the cardiovascular health of persons in the 30–50 age range? Null and Alternative Hypotheses: ●​ Null Hypothesis (H₀): When adults between the ages of 30 and 50 participate in a physical fitness intervention, there is no discernible change in their resting heart rate at baseline, two months, or four months. ●​ Alternative Hypothesis (H₁): When adults between the ages of 30 and 50 participate in a physical fitness intervention, their resting heart rates change significantly over time (baseline, two months, and four months). Effect Size: The impact size in this study will quantify how much the resting heart rate differs at baseline, two months, and four months. Cohen's d or partial eta squared for repeated measures may be used as particular estimates of effect size. The degree to which the physical fitness intervention is genuinely lowering resting heart rate at each of the three time periods will be ascertained with the use of the impact size. Whereas a smaller effect size would imply a weaker or nonexistent effect, a bigger effect size would indicate a stronger influence of the intervention on cardiovascular health. Type 1 Error: ●​ Definition: A Type 1 error happens when the null hypothesis is wrongly rejected, leading us to believe that the intervention has a large impact on resting heart rate when in reality there isn't. ●​ Implication in this study: If there is a Type 1 error, we incorrectly believe that the physical fitness intervention had a significant long-term improvement in participants' cardiovascular health (resting heart rate), whereas in fact there was no such effect. This can result in erroneous conclusions about the efficacy of the intervention and even needless suggestions or policies. Type 2 Error: ●​ Definition: When the null hypothesis is mistakenly maintained, we fail to identify a significant effect of the intervention when, in reality, there is a true effect. This is known as a Type 2 error. ●​ Implication in this study: A Type 2 mistake would indicate that, despite the physical fitness intervention's effectiveness, we were unable to identify that it had a long-term positive impact on resting heart rate. This can cause the study to overlook the intervention's genuine benefits, leading us to draw the wrong conclusion that it isn't beneficial when in fact it is. Summary of Errors: ● Type 1 error: concluding that, when it doesn't, the physical fitness intervention dramatically raises resting heart rate. ● Type 2 error: failing to recognize when there is a real improvement in resting heart rate as a result of the intervention. REFERENCES: ● Illowsky, B. and Dean, S. (2017). Introductory Statistics. OpenStax Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).