Hypothesized That Girls In My Class Have The Same Blood Pres ✓ Solved

Hypothesized That Girls In My Class Have The Same Blood Pressure Lev

Analyze the scenario where a student hypothesized that girls in their class have the same blood pressure levels as boys, with a null hypothesis probability value of 0.15. The student concluded that the blood pressures of girls were higher than boys, which was identified as an error. Determine what type of mistake was made, whether a Type I error, Type II error, both, or if no mistake was made.

Additionally, interpret the meaning of a non-significant result at the 0.05 level, and what implications it has for significance at different levels, such as 0.10 or 0.01.

In a related testing scenario, a researcher tested whether there was a significant difference between boys' and girls' memory scores. The sample included 100 girls with a mean score of 70 and a standard deviation of 5, and 100 boys with a mean score of 65 and the same standard deviation. Calculate the standard error of the difference between means.

Furthermore, discuss the implications of high statistical power, specifically how increased power impacts the likelihood of detecting true effects and the relationship with significance levels and errors.

Consider a scenario where the probability of correctly rejecting a false null hypothesis (statistical power) is 0.80 at a significance level of 0.05. Clarify what this indicates about α (the significance level) and β (the Type II error rate).

Using a population with a mean of 678 and a known standard deviation of 58.3, perform a hypothesis test at α = 0.05 to assess whether a sample's scores originate from this population. Provide the hypotheses, test statistic, P-value or critical value, and conclusion regarding the null hypothesis and the original claim.

Finally, evaluate a claim that the average house prices in Mississippi are greater than in Arkansas. Given a random sample of 12 houses from each state with respective costs, conduct a significance test at α = 0.05 to compare the means, providing hypotheses, test statistic with P-value, and a conclusive interpretation.

Sample Paper For Above instruction

Introduction

Statistical hypothesis testing is a crucial method in research for determining whether observed data supports or refutes a presumed assumption, called the null hypothesis. The correct identification and understanding of the types of errors involved, as well as the implications of significance levels and statistical power, are fundamental to accurate interpretation of results. This paper explores several scenarios involving hypothesis testing, error types, power, and real-world applications such as blood pressure comparison, memory scores, and housing costs.

Error Identification in Blood Pressure Scenario

The scenario involves a student hypothesizing that girls share the same blood pressure levels as boys with a null hypothesis probability of 0.15. The student concludes that girls have higher blood pressure than boys, but this leads to a misinterpretation of the statistical results. The key issue is the type of error committed.

A Type I error occurs when a true null hypothesis is incorrectly rejected, whereas a Type II error involves failing to reject a false null hypothesis. In this case, the student’s conclusion that girls have higher blood pressure, despite a p-value of 0.15 (which is greater than the common significance threshold of 0.05), indicates a misinterpretation leading to a false positive in real effect. This constitutes a Type I error, as the students incorrectly rejected the null hypothesis of equality when the evidence was insufficient to do so.

Significance at Different Levels

If a statistical test result is not significant at the 0.05 level, it does not necessarily imply a lack of significance at a higher threshold such as 0.10. Conversely, it also implies the result is not significant at the 0.01 level. Specifically, a non-significant at 0.05 signifies p > 0.05; it may be either significant or not at 0.10 depending on the p-value, but it is definitively not significant at the 0.01 level since 0.01 is more stringent. Thus, the statement "It is not significant at 0.10 level" is not necessarily true, but "It is not significant at 0.01 level" is accurate.

Comparison of Memory Scores Between Boys and Girls

The study involves comparing mean memory scores for two groups: 100 girls with a mean of 70 and a standard deviation of 5, and 100 boys with a mean of 65 and the same standard deviation. To compare these, the standard error of the difference in means must be calculated.

The standard error (SE) of the difference is given by: SE = sqrt((SD1^2 / n1) + (SD2^2 / n2)). Here, SD1 = SD2 = 5, n1 = n2 = 100.

Calculating: SE = sqrt((25 / 100) + (25 / 100)) = sqrt(0.25 + 0.25) = sqrt(0.5) ≈ 0.7071.

The Impact of Power in Hypothesis Testing

Power refers to the probability of correctly rejecting a false null hypothesis, i.e., detecting a true effect. When power is high, the test is more sensitive, and true differences are more likely to be recognized. Increasing power can be achieved by increasing sample size, using more precise measurements, or choosing a less stringent significance level. High power reduces the likelihood of Type II errors, making results more conclusive and trustworthy.

Interpreting Power and Error Rates

Given that the probability of correctly rejecting a false null hypothesis (power) is 0.80 at α = 0.05, this indicates that the significance level α is 0.05, and the probability of a Type II error (β) is 0.20. This relationship aligns with the standard framework of hypothesis testing, where increasing power (reducing β) improves the likelihood of detecting real effects.

Population Mean Testing

Testing whether a sample's scores come from a population with a known mean of 678 and standard deviation of 58.3 involves calculating the z-test statistic: z = (sample mean - population mean) / (population standard deviation / sqrt(n)). The critical value for a two-tailed test at α = 0.05 is approximately ±1.96.

The p-value can be derived from the z-score or compared with the critical value to determine significance. The conclusion depends on whether z falls outside the critical bounds and whether the p-value is below 0.05, indicating whether to reject the null hypothesis.

Housing Cost Comparison Between Mississippi and Arkansas

The last scenario involves comparing the mean house costs in Mississippi and Arkansas based on random samples. The hypotheses are: H0: μ_Mississippi ≥ μ_Arkansas; HA: μ_Mississippi > μ_Arkansas. A two-sample t-test (assuming equal variances) is conducted with the sample means, standard deviations, and sample sizes.

Calculate the test statistic (t), determine the P-value, and interpret whether the data supports the claim that Mississippi houses are more expensive than Arkansas houses at the significance level of 0.05.

Conclusion

Hypothesis testing involves understanding the types of errors, the role of significance levels, statistical power, and proper interpretation of outcomes. These examples highlight the importance of precise calculations and cautious conclusions in research to avoid erroneous inferences, ultimately strengthening the reliability of scientific findings.

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