You Must Show Your Work: Math Teacher Believes That She Has

You Must Show Your Work1a Math Teacher Believes That She Has Develope

The assignment involves formulating hypotheses and conducting appropriate statistical tests based on provided data. Specifically, the tasks include testing the impact of a review course on students' SAT scores, evaluating hippocampal volume differences among adolescents with alcohol use disorders, and assessing whether satisfaction with promotion chances among employed adults has changed over time. The essence of the assignment is to perform hypothesis testing at specified significance levels using sample data and interpret the results accordingly.

Paper For Above instruction

Introduction

Statistical hypothesis testing is a fundamental method in research for determining whether there is enough evidence to support a specific claim about a population parameter. It involves formulating null and alternative hypotheses, selecting an appropriate test, calculating the test statistic, and making a decision based on the significance level (α). This paper analyzes three different scenarios where hypothesis testing is applied to assess the effectiveness of a review course, the impact of alcohol on adolescent hippocampal volume, and the change in employee satisfaction regarding promotion prospects.

Scenario 1: Impact of a Review Course on SAT Scores

The first scenario examines whether a review course improves students' scores on the SAT math section. The data shows that, prior to the course, scores are normally distributed with a mean (μ) of 523 and a standard deviation (σ) of 113. After conducting the course on a sample of 1800 students, the class averages increased slightly to 526.

Hypotheses Formulation:

  • Null hypothesis (H₀): The review course has no effect on SAT scores; therefore, μ = 523.
  • Alternative hypothesis (H₁): The review course increases SAT scores; therefore, μ > 523.

This is a one-tailed z-test for the population mean, considering the large sample size and known population standard deviation. The test statistic (z) is calculated as:

z = (x̄ - μ₀) / (σ / √n) = (526 - 523) / (113 / √1800)

Calculating:

  • Standard Error (SE) = 113 / √1800 ≈ 113 / 42.43 ≈ 2.66
  • Z = (3) / 2.66 ≈ 1.13

Using standard normal distribution tables or software, the p-value corresponding to z = 1.13 (one-tailed) is approximately 0.1292.

Decision: Since p = 0.1292 > α = 0.10, we fail to reject the null hypothesis. There is insufficient evidence at the 10% significance level to conclude that the review course increases SAT scores.

Scenario 2: Effect of Alcohol on Hippocampal Volume in Adolescents

The second scenario investigates whether adolescents with alcohol use disorder have smaller hippocampal volumes than the typical volume of 9.02 cm³. The sample involves 11 adolescents with mean hippocampal volume of 8.49 cm³ and standard deviation 0.8 cm³, assuming normality.

Hypotheses:

  • H₀: The mean hippocampal volume is equal to the normal volume, μ = 9.02 cm³.
  • H₁: The mean hippocampal volume is less than 9.02 cm³, μ

This is a one-sample t-test for the mean, given the small sample size (n

Test statistic:

t = (x̄ - μ₀) / (s / √n) = (8.49 - 9.02) / (0.8 / √11)

Calculating:

  • Standard Error (SE) = 0.8 / √11 ≈ 0.8 / 3.3166 ≈ 0.241
  • t = (-0.53) / 0.241 ≈ -2.20

Degrees of freedom: df = n - 1 = 10. Looking up the critical t-value for a one-tailed test at α = 0.01 and df = 10 yields approximately -2.764.

Decision: Our calculated t (-2.20) does not exceed the critical t-value (-2.764). Since -2.20 > -2.764, we fail to reject the null hypothesis at the 1% significance level. There is not enough evidence to conclude that the hippocampal volumes in alcoholic adolescents are smaller than the normal volume.

Scenario 3: Change in Employee Satisfaction with Promotion Opportunities

In the third scenario, the goal is to test if the proportion of employed adults satisfied with their chances for promotion has changed since a previous survey that reported 21% satisfaction. A random sample of 260 adults finds 65 satisfied individuals.

Hypotheses:

  • H₀: The proportion remains at 21%, p = 0.21.
  • H₁: The proportion has changed, p ≠ 0.21.

This is a two-proportion z-test for a population proportion. The sample proportion:

p̂ = 65 / 260 = 0.25

Standard error:

SE = √[p₀(1 - p₀) / n] = √[0.21 * 0.79 / 260] ≈ √(0.1659 / 260) ≈ √(0.000638) ≈ 0.0252

Test statistic:

z = (p̂ - p₀) / SE = (0.25 - 0.21) / 0.0252 ≈ 0.04 / 0.0252 ≈ 1.59

Critical value at α = 0.1 for a two-tailed test: approximately ±1.645. Since |z| = 1.59

Conclusion: There is not sufficient evidence to suggest that the proportion of satisfied adults significantly differs from 21% at the 10% significance level.

Conclusion

In all three scenarios, hypothesis testing provides crucial insights into the data. The first scenario indicates insufficient evidence to confirm that the review course improves SAT scores at the 10% significance level. The second scenario suggests that, with high confidence, hippocampal volumes in adolescents with alcohol use disorder are not significantly smaller than normal at the 1% level. The third scenario reveals no statistically significant change in the proportion of satisfied adults regarding promotions at the 10% significance level. These analyses underscore the importance of choosing appropriate tests, understanding significance levels, and accurately interpreting statistical results in research contexts.

References

  • Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences (10th ed.). Cengage Learning.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage Publications.
  • Moore, D. S., Notz, W., & Fligner, M. A. (2018). The Basic Practice of Statistics (8th ed.). W. H. Freeman.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Taylor, R. (2014). Introductory Statistics (7th ed.). Pearson.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks Cole.
  • Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses (3rd ed.). Springer.
  • Greenwood, P. E., & Durand, D. (2012). Statistics for +Business & Economics (3rd ed.). Oxford University Press.
  • Chen, M. H. (2020). Using R for Introductory and Advanced Hypothesis Tests. Chapman & Hall/CRC.
  • McClave, J. T., & Sincich, T. (2018). Statistics (13th ed.). Pearson.

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