ABC/123 Version X 1: Probability And Statistical Analysis

ABC/123 Version X 1 Probability and Statistical Analysis Worksheet

Complete Parts A, B, and C below.

Part A:

1. Why is a z score a standard score? Why can standard scores be used to compare scores from different distributions? Why is it useful to compare different distributions?

2. For the following set of scores, fill in the cells. The mean is 74.13 and the standard deviation is 9.98. Raw score Z score 68.0 ? ? –1..0 ? ? 1..0 ? ? –0..0 ? ? 1..0 ?

3. Questions 3a through 3d are based on a distribution of scores with a standard deviation of 6.38. Draw a small picture to help you see what is required.

a. What is the probability of a score falling between a raw score of 70 and 80?

b. What is the probability of a score falling above a raw score of 80?

c. What is the probability of a score falling between a raw score of 81 and 83?

d. What is the probability of a score falling below a raw score of 63?

4. Jake needs to score in the top 10% in order to earn a physical fitness certificate. The class mean is 78 and the standard deviation is 5.5. What raw score does he need?

Part B:

The questions in Part B require that you access data from the Pulse Rate Dataset. The data is based on the following research problem: Ann conducted a study on the factors affecting pulse rate after exercising. She wants to describe the demographic characteristics of a sample of 55 individuals who completed a large-scale survey. She has demographic data on gender (two categories), age (open-ended), level of exercise (three categories), height (open-ended), and weight (open-ended).

5. Using Microsoft® Excel®, run descriptive statistics on gender and level of exercise variables. From the output, identify:

a. Percent of men

b. Mode for exercise frequency

c. Frequency of high-level exercisers (exercise level 1)

6. Using Microsoft® Excel®, run descriptive statistics to summarize the age variable, noting the mean and standard deviation. Copy and paste the output into this worksheet.

Part C:

Answer the questions below in at least 90 words. Be specific and provide examples when relevant. Cite any sources according to APA guidelines.

- How does understanding probability help you understand inferential statistics? When have you used probability in everyday life? How did you use it?

- Which do you think would be a more serious violation: a Type I or Type II error? And why?

- What are the characteristics that separate parametric and nonparametric tests?

- What does statistical significance mean? How do you know if something is statistically significant?

- What is the difference between statistical significance and practical significance?

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Paper For Above instruction

Understanding Probability and Its Role in Inferential Statistics

Probability plays a fundamental role in understanding inferential statistics, which involves making predictions or inferences about a population based on sample data. At its core, probability quantifies the likelihood that a specific event will occur, providing the foundation for statistical inference. For example, when a researcher wants to determine whether a new medication is effective, they rely on probabilities derived from sample data to infer the effect on the entire population. Without an understanding of probability, interpreting these results, such as p-values or confidence intervals, would be impossible, as it enables us to measure the uncertainty inherent in sample data.

In everyday life, probability is used in various contexts. For instance, weather forecasts predict the likelihood of rain, guiding our decisions on outdoor activities. When buying insurance, individuals assess the probability of certain events occurring to determine premiums. Similarly, in sports, analysts estimate the chances of a team winning based on historical data. These experiences demonstrate how probability helps us gauge uncertainty and make informed decisions, reflecting its vital role in both practical and scientific realms.

Seriousness of Type I and Type II Errors

In statistical hypothesis testing, a Type I error occurs when a true null hypothesis is incorrectly rejected (a false positive), whereas a Type II error happens when a false null hypothesis fails to be rejected (a false negative). Generally, a Type I error is considered more serious because it can lead to the acceptance of a false claim, such as concluding a new drug is effective when it is not, which can have significant ethical and health consequences. Conversely, Type II errors might result in missing out on beneficial discoveries, but the societal impact of falsely claiming effectiveness is often deemed more harmful. Therefore, controlling the risk of Type I errors is usually prioritized, especially in medical and scientific research.

Differences Between Parametric and Nonparametric Tests

Parametric tests are statistical procedures that assume underlying data distributions follow specific parameters, typically a normal distribution. They are appropriate when data meet assumptions such as homogeneity of variances and interval or ratio scale measurement. Examples include t-tests and ANOVA. Nonparametric tests, on the other hand, do not assume a specific distribution and are used when data violate parametric assumptions or are ordinal or nominal. Examples include the Mann-Whitney U test and Chi-square test. The main characteristic separating these tests is the distribution requirement: parametric tests assume a known distribution, whereas nonparametric tests do not, making the latter more flexible but often less powerful.

Statistical Significance and Its Implications

Statistical significance indicates that an observed effect or relationship is unlikely to have occurred by chance alone, given a predetermined alpha level (commonly 0.05). When a p-value falls below this threshold, we infer that the results are statistically significant, meaning there is enough evidence to reject the null hypothesis. However, statistical significance does not necessarily imply practical importance. A result can be statistically significant but have minimal real-world impact, especially in large samples where even trivial differences can be detected. Therefore, researchers must consider both statistical and practical significance to make meaningful conclusions about their data.

Distinguishing Statistical and Practical Significance

While statistical significance pertains to the likelihood that a result is not due to chance, practical significance assesses whether the effect size or difference is meaningful in real-world terms. For example, a new teaching method may statistically improve test scores, but if the improvement is only a fraction of a point, it might not justify implementation due to practical considerations such as cost or feasibility. Conversely, a practically significant result might be clinically meaningful even if it does not reach statistical significance in small samples. Balancing these aspects is crucial for responsible research, ensuring findings are both reliable and relevant.

References

  • Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Routledge.
  • Field, A. (2013). Discovering statistics using IBM SPSS Statistics. Sage Publications.
  • Gravetter, F. J., & Wallnau, L. B. (2018). Statistics for the behavioral sciences (10th ed.). Cengage Learning.
  • Hogg, R. V., McKean, J., & Craig, A. T. (2019). Introduction to mathematical statistics (8th ed.). Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the practice of statistics (9th ed.). W.H. Freeman.
  • Myers, J. L., Well, A. D., & Lorch, R. F. (2010). Research design and statistical analysis. Routledge.
  • Rice, J. (2007). Mathematical statistics and data analysis. Cengage Learning.
  • Statistical Package for the Social Sciences (SPSS). (2020). IBM Corporation.
  • Urdan, T. C. (2016). Statistics in plain English (4th ed.). Routledge.
  • Wasserman, L. (2004). All of statistics: A concise course in statistical inference. Springer.

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