Deliverable 5: Hypothesis Tests For Two Samples ✓ Solved

Deliverable 5 Hypothesis Tests For Two Samples

Formulate and evaluate hypothesis tests for population parameters based on sample statistics using both Critical Regions and P-Values, and be able to state results in a non-technical way that can be understood by consumers of the data instead of statisticians. Dealing with Two Populations Inferential statistics involves forming conclusions about a population parameter. We do so by constructing confidence intervals and testing claims about a population mean and other statistics. Typically, these methods deal with a sample from one population. We can extend the methods to situations involving two populations (and there are many such applications).

This deliverable looks at two scenarios. You are asked to demonstrate your understanding of hypothesis testing for two populations, including independent and dependent samples. Your work should include detailed calculations, clear explanations of each step, and non-technical interpretations of the results. All research, calculations, and analysis should be presented on the provided spreadsheet template, with all red text items deleted before submission.

Sample Paper For Above instruction

The purpose of this paper is to demonstrate the process of conducting hypothesis tests for two population samples, considering both independent and dependent samples, with detailed steps, calculations, and interpretative conclusions suitable for a non-technical audience. The focus is to apply statistical methods properly, interpret results accurately, and communicate findings clearly.

Introduction

Hypothesis testing is a fundamental statistical technique used to make inferences about population parameters based on sample data. When dealing with two populations, the analysis becomes more complex, involving comparisons between two groups, either independent or dependent. The purpose of this paper is to illustrate how to perform such tests, interpret the results, and communicate findings effectively.

Part 1: Hypothesis Testing for Independent Samples

Suppose a researcher is interested in comparing the IQ scores of individuals with low and high lead levels in their blood. Two independent samples are taken from these populations. The sample from the low lead group has a size of n₁=78 with a standard deviation of s₁=15.34, while the high lead group has n₂=..9 (assumed 78 for illustration) with standard deviation s₂=8.99. The goal is to test whether the mean IQ score of the low lead group is higher than that of the high lead group, using a significance level of 0.05.

Step 1 involves stating hypotheses:

• Null hypothesis (H₀): μ₁ ≤ μ₂ (mean IQ of low lead ≤ mean IQ of high lead)
• Alternative hypothesis (H₁): μ₁ > μ₂ (mean IQ of low lead > mean IQ of high lead)

This is a right-tailed test since we are testing whether the first mean is greater than the second.

Step 2 is calculating the test statistic:

Using the formula for two independent sample t-test with unequal variances (Welch's t-test):

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

Based on sample data, suppose:

• Sample means: x̄₁= (assumed value)
• Sample means: x̄₂= (assumed value)
• Calculations proceed accordingly to compute t, degrees of freedom, and the corresponding p-value.

The critical value is determined from t-distribution tables or software for df estimated via Welch's formula.

Step 3 involves making a decision: if the calculated p-value is less than 0.05, reject H₀, concluding that there is evidence that the mean IQ of the low lead group is higher. Otherwise, do not reject H₀, indicating insufficient evidence.

Part 2: Hypothesis Testing for Dependent Samples

Consider a study examining the effect of a treatment on days taken to release a book. Paired data is collected for two movies, Phoenix and Prince. The differences in days are analyzed with n=5 pairs, with differences in days noted for each pair.

Step 1 is to state hypotheses:

• H₀: The mean difference in days between the two movies = 0
• H₁: The mean difference ≠ 0 (two-tailed test)

Step 2 involves calculating the mean difference, standard deviation, and then the t-statistic:

\[ t = \frac{\bar{d}}{s_d/\sqrt{n}} \]

Compare the t-statistic to the critical t-value for n-1 degrees of freedom at 0.05 significance level. Calculate the p-value accordingly.

If p-value

Part 3: Interpreting Results and Communicating Findings

It is essential to communicate the results in plain language. For example, if the test shows a significant difference, one might say: "There is enough evidence to suggest that the release durations for the two movies are different." If not, then: "The data does not provide sufficient evidence to conclude a difference exists."

Conclusion

Hypothesis testing for two samples enables researchers to compare groups effectively. Proper formulation of hypotheses, accurate calculation of test statistics, and correct interpretation of p-values are critical steps. Clear, non-technical communication ensures that findings are accessible and meaningful for decision-makers and the general public.

References

• Laerd Statistics. (2023). Two independent samples t-test in Practice. Retrieved from https://statistics.laerd.com
• Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics. SAGE Publications.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics. W.H. Freeman.
• Ott, R. L., & Longnecker, M. (2015). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Gravetter, F. J., & Wallnau, L. B. (2016). Statistics for Behavioral Science. Cengage Learning.
• Wasserstein, R. L., & Lazar, N. A. (2016). The ASA's Statement on p-Values: Context, Process, and Purpose. The American Statistician.
• Chow, S.-C., & Liu, J. P. (2013). Design and Analysis of Clinical Trials. CRC Press.
• Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern Epidemiology. Lippincott Williams & Wilkins.
• Steven, G. (2013). Introductory Statistics. Pearson.
• McClave, J. T., & Sincich, T. (2015). A First Course in Statistics. Pearson.