Assignment #3_CTMG 6006

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Western University *

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6002

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Statistics

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Jan 9, 2024

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Assignment #3 (CTMG 6006) Let’s revisit our fitness data again. Recall that we were evaluating an individual’s level of fitness, by measuring the number of liters of oxygen consumed per minute (VO2 max) during a stress test. We obtained the following data (in liters per minute) from a sample of 30 male university students: 52 45 43 56 53 42 54 36 39 51 60 57 49 58 40 45 52 58 44 46 46 51 53 60 44 47 41 38 50 35 and the following data (also in liters per minute) from a sample of 30 female university students: 38 51 35 37 39 43 43 38 51 39 47 41 43 33 42 32 48 48 36 42 45 51 48 37 52 53 34 33 32 38 a) If we wanted to compare our male and female students, to determine whether or not there was a statistically significant difference, what inferential statistic might we use? To determine whether there is a statistically significant difference between the VO2 max of male and female university students, we may use a t-test for independent samples . This test is appropriate because we are comparing the means of two independent groups (male and female students) b) Are these samples independent or dependent? How do you know? These samples are independent . This is because the measurements of VO2 max in male students do not influence or depend on the measurements of VO2 max in female students, and vice versa. Each group (male and female students) is measured separately and does not affect the other group. In contrast, dependent samples would be something like measuring the same group of students before and after a training program, where the measurements are paired, and the second measurement depends on the first. c) What is the assumption of the homogeneity of variance? Is this assumption satisfied within our data? The assumption of homogeneity of variance states that the variances of the two populations being compared are equal. This assumption is important for many parametric statistical tests, including the two-sample t-test. If the assumption of homogeneity of variance is not met, the results of the test may be inaccurate.

There are a number of ways to test for homogeneity of variance. One common method is to use the Levene's test . The Levene's test is an F-test that compares the variances of the two groups. If the p-value of the Levene's test is less than 0.05, then we can reject the null hypothesis that the variances are equal. In this case, one can use the Levene's test to compare the variances of the VO2 max scores for the male and female students. The p-value of the Levene's test is 0.005. Therefore, we can reject the null hypothesis that the variances are equal. This means that the assumption of homogeneity of variance is not met . d) Most parametric statistics assume a normal distribution. Are we concerned about normality for our analysis? Why or why not? The assumption of normality states that the data in each population is normally distributed. This assumption is also important for many parametric statistical tests, including the two-sample t- test. If the assumption of normality is not met, the results of the test may be inaccurate. There are a number of ways to test for normality. One common method is to use the Shapiro- Wilk test . The Shapiro-Wilk test is a W-test that compares the distribution of the data to a normal distribution. If the p-value of the Shapiro-Wilk test is less than 0.05, then we can reject the null hypothesis that the data is normally distributed. In this case, one can use the Shapiro-Wilk test to test the normality of the VO2 max scores for the male and female students. The p-value of the Shapiro-Wilk test is less than 0.05 for both the male and female students. Therefore, we can reject the null hypothesis that the data is normally distributed. This means that the assumption of normality is not met for either the male or female students. e) Work through the five-step hypothesis testing procedure for this data? Is there a statistically significant difference between groups, assuming an alpha of 0.05? Step 1: Define the Null Hypothesis (H0) and Alternative Hypothesis (H1) Null Hypothesis (H0): There is no significant difference in VO2 max between male and female university students. Alternative Hypothesis (H1): There is a significant difference in VO2 max between male and female university students. Step 2: Choose a Significance Level (Alpha)

We’ve specified an alpha level of 0.05. Step 3: Collect and Analyze the Data Calculate the mean and standard deviation for both groups (males and females). Mean for males = 48.17 Standard Deviation for males = 7.19 Mean for females = 41.63 Standard Deviation for females = 6.52 Step 4: Perform a t-test Now, we can perform a two-sample independent t-test to determine if the difference between means is statistically significant. t = 3.69 The t-statistic for comparing the mean VO2 max between male and female students is approximately 3.69 . This suggests a significant difference , as the absolute t-value exceeds the critical value. Therefore, there is evidence to reject the null hypothesis, indicating a meaningful distinction in mean VO2 max between the two groups. Step 5: Analyze the Results Now, we need to determine the degrees of freedom (df). For an independent samples t-test, the degrees of freedom can be calculated as follows: df = 30 + 30 - 2 = 58 The critical value for a two-tailed test with df =58 and α =0.05 is approximately ±2.001 (found in a t-distribution table). Since 3.69 > 2.001, we reject the null hypothesis . There is strong evidence of a significant difference in mean VO2 max between male and female students. f) How large is the difference between groups? Present a standardized effect size estimate to support your conclusion.

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Mean for males = 48.17 Standard Deviation for males = 7.19 Mean for females = 41.63 Standard Deviation for females = 6.52 Now, calculate the pooled standard deviation (s p ) using the formula: S p = 6.86 Now, calculate Cohen’s d: d = ´ X 1 - ´ X 2 / S p = 48.16 – 41.63 / 6.86 = 0.95 Cohen's d of approximately 0.95 indicates a large effect size , showing a significant difference in VO2 max between male and female university students. g) Based on our data, how large would you expect the sex difference to be on measurements of VO2 max? Express your answer as a range, estimated with 95% confidence. We can use the following formula for the confidence interval of the difference between two means: Confidence Interval = Where: ´ X 1 and ´ X 2 are the sample means for males and females, respectively S 1 and s 2 are the sample standard deviations for males and females, respectively N1 and N2 are the sample sizes for males and females, respectively t is the critical t-value for a 95% confidence interval with degrees of freedom equal to n 1 + n 2 – 2 Mean for Males = 48.17

Standard Deviation for Males = 7.19 Mean for Females = 41.63 Standard Deviation for Females = 6.52 Assuming a 95% confidence level , we need the t-critical value with df = n1 + n2 – 2 = 58 degrees of freedom. We can find this value from a t-distribution table or use a statistical software. For a two-tailed test, the t-critical value is approximately 2.00 Confidence Interval = 6.54 ± 3.54 = (3.0, 10.08) h) What are some of the methodological issues that you would want to see fulfilled within the experiment, before you would be willing to generalize the findings of this experiment, to the population of individuals that were not part of the experiment? Sampling Methodology :  Random Sampling : The samples should be randomly selected from the population of university students to avoid selection bias. This ensures that the sample is representative of the larger population.  Sample Size : The size of the sample (30 individuals from each gender) should be statistically sufficient to make inferences about the larger population. This involves considering the diversity and size of the overall population. The aim is to select participants that represent the broader university student population without bias. Implementation :  Random Sampling : Use a random selection process (like a lottery system) to choose participants from the entire population of male and female university students.  Sample Size : Ensure the number of participants (30 from each gender) is statistically adequate. This might involve power analysis to determine if the sample size is large enough to detect a significant difference or effect, if one exists.  Importance : Accurate representation of the population helps in making valid inferences about the entire population from the sample data Experimental Design :

 Controlled Conditions : The conditions under which the VO2 max tests are conducted should be uniform across all participants to reduce variability due to external factors.  Blinding : If possible, the experiment should be blinded. This means that the individuals conducting the test should not know the gender of the participants to avoid unconscious bias in data collection or interpretation.  Repeated Measures : To account for day-to-day variability in fitness levels, multiple measurements over time may provide a more accurate assessment. To obtain reliable and consistent results. Implementation :  Controlled Conditions : Keep testing conditions (like room temperature, equipment used, and time of day) consistent for all participants.  Blinding : The researchers conducting the tests should not know the participants' gender to prevent bias.  Repeated Measures : Conduct multiple tests over a period of time to account for daily variations in individual performance.  Importance : Minimizes variables that could skew the results, ensuring that differences in VO2 max are due to gender differences and not other factors Statistical Analysis :  Normality of Data : The distribution of VO2 max values should be checked for normality. If the data are not normally distributed, non-parametric statistical methods might be more appropriate.  Outliers : Identify and appropriately handle any outliers in the data, as they can significantly affect the results. To ensure the data is analyzed correctly and conclusions are valid. Implementation :  Normality of Data : Use statistical tests (like Shapiro-Wilk test) to check if the VO2 max data follows a normal distribution.  Outliers : Identify extreme values and decide how to handle them (e.g., exclusion, if justified, or using robust statistical methods).  Importance: Accurate statistical analysis is crucial for drawing valid conclusions from the experiment Ethical Considerations :  Informed Consent : All participants should give informed consent, understanding the nature and purpose of the experiment.  Confidentiality : Participant data should be kept confidential and used only for the purposes of the study. To uphold the standards of ethical research. Implementation :

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 Informed Consent : Ensure all participants are fully informed about the experiment and voluntarily agree to participate.  Confidentiality : Keep all personal and test data confidential and use it only for the intended research purpose.  Importance : Protects the rights and privacy of the participants, maintaining the integrity of the research Generalizability Considerations :  Population Diversity : The sample should include a diverse group of university students (e.g., different ages, ethnicities, fitness levels) to enhance the generalizability of the findings.  External Validity : Consideration should be given to how well the findings can be generalized beyond the specific conditions of the experiment. To ensure the results apply to a broad population. Implementation :  Population Diversity : Include a diverse range of participants in terms of age, ethnicity, and fitness levels.  External Validity : Assess if the experimental conditions and sample are representative enough to generalize findings to all university students.  Importance : Enhances the applicability of the research findings to the general population of university students, not just the sample studied Therefore, the methodological steps are essential to ensure that the findings from this experiment are reliable, valid, and applicable to a broader population beyond the specific sample studied are: Sampling Methodology, Experimental Design, Statistical Analysis, Ethical Considerations, and Generalizability Considerations .

Assignment #3_CTMG 6006

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