Application: Hypothesis Testing: Making Inferences from a Sample ✓ Solved

Hypothesis testing is the foundation of conducting research in psychology. Researchers must first determine the question they wish to answer and then state their prediction in terms of null and alternative hypotheses. After stating the hypotheses and collecting data, the researcher needs to determine if the results are statistically significant. For instance, a researcher may ask whether a drug for severe depression affects life expectancy compared to the general population of people with severe depression.

In this case, a researcher investigates if eighth-grade students attending a private middle school have higher or lower scores on a reading comprehension test compared to eighth-graders in public schools. A sample of 144 private school eighth-graders has a mean test score of 220.8, while the mean score for public school students is 204.2, with a standard deviation of 11.4.

To complete this Assignment, respond to the following:

  • State the independent and dependent variables and explain how you know which is which.

  • Explain whether the researcher should use a one-tailed or a two-tailed z test and explain why.

  • State the null hypothesis in words (not formulas).

  • State the alternative hypothesis in words (not formulas).

  • Calculate the obtained z score by hand. Provide your calculations.

  • When alpha is set at .05, the critical value is ± 1.96. Would you retain or reject the null hypothesis? Explain why?

  • Are the results significant? How do you know?

  • What should the researcher conclude about the reading comprehension of the private school students in comparison to the population?

  • In general, explain the relationship between z scores and the standard deviation. Justify your responses with evidence from the text and learning resources.

Provide an APA reference list.

Reference to use: Week 3 Learning Resources.

Paper For Above Instructions

Hypothesis testing is a fundamental component of statistical analysis within psychology and other disciplines, allowing researchers to make inferences about populations based on sample data. In addressing this assignment, we will explore various elements necessary for conducting hypothesis testing, particularly in the context of the research scenario outlined.

Independent and Dependent Variables

The independent variable in this scenario is the type of school attended by the eighth-grade students: private versus public (Heiman, 2015). The dependent variable is the students' scores on the reading comprehension test. This distinction is evident because the independent variable (type of school) is manipulated to observe its effect on the dependent variable (test score). The basis for identifying these variables lies in the causal relationship where changes in the independent variable lead to variations in the dependent variable.

Choice of Z-Test

In this situation, the researcher should employ a one-tailed z-test. The rationale behind this choice is based on the hypothesis that private school students will outperform public school students on the reading comprehension test. Since there is a specific directional expectation (higher scores), a one-tailed test is appropriate as it allows us to detect this difference (Heiman, 2015). Conversely, a two-tailed z-test would be used if we were interested in any difference, regardless of direction.

Null and Alternative Hypotheses

The null hypothesis (H0) can be stated as follows: There is no difference in the reading comprehension test scores between eighth-grade students attending private middle schools and those attending public schools. Conversely, the alternative hypothesis (H1) states: Eighth-grade students attending private middle schools have higher reading comprehension test scores as compared to those attending public schools.

Calculation of the Obtained Z Score

To calculate the z-score, we will first determine the mean score differences and then apply the z-score formula. The mean score of private school students is \( \bar{X}_{private} = 220.8 \) and of public school students \( \bar{X}_{public} = 204.2 \). The formula for the z-score in this scenario is:

Z-score formula

Our standard error (SE) can be calculated using the formula: SE = SD / √n where SD is the standard deviation (11.4) and n is the sample size (144).

Calculating SE: SE = 11.4 / √144 = 11.4 / 12 = 0.95.

Next, we compute the z-score: z = (X̄1 - X̄2) / SE z = (220.8 - 204.2) / 0.95 z = 16.6 / 0.95 = 17.5.

Decision on the Null Hypothesis

Given that the calculated z-score (17.5) far exceeds the critical z-value of ±1.96 at a significance level of alpha = 0.05, we reject the null hypothesis. This result suggests that there is a statistically significant difference in the reading comprehension scores between private and public school eighth-graders.

Significance of the Results

The results are significant as evidenced by the rejection of the null hypothesis. The extremely high z-score indicates that the mean differences between the two groups are large enough to be unlikely to have occurred by random chance (Heiman, 2015).

Conclusion About Reading Comprehension

The researcher can conclude that eighth-grade students in private schools perform significantly better in reading comprehension tests compared to their peers in public schools. This finding could point to differences in educational strategies, resources available, or other contextual factors that merit further investigation.

Relationship Between Z Scores and Standard Deviation

The relationship between z-scores and standard deviation is critical in understanding how individual data points relate to the overall distribution. A z-score indicates how many standard deviations an observation is from the mean. In this scenario, a positive z-score suggests that the reading comprehension scores of private school students are above average compared to public school students, highlighting the effectiveness of their educational environment (Heiman, 2015).

References

  • Heiman, G. (2015). Behavioral sciences STAT (2nd ed.). Stamford, CT: Cengage.

  • Laureate Education. (2013b). Introduction to hypothesis testing [Video file].

  • StatisticsLectures.com. (2012d). Z-scores [Video file].

  • Khan Academy. (2013a). Introduction to normal distribution [Video file].

  • Texas A & M University. (n.d.). Psychic test.

  • StatisticsLectures.com. (2012c). Type I and II errors [Video file].

  • StatisticsLectures.com. (2012a). Null and alternative hypotheses [Video file].

  • Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). Sage Publications.

  • Peck, R. J., & Olsen, C. (2015). Introduction to Statistics and Data Analysis. Cengage Learning.

  • Gravetter, F. J., & Wallnau, L. B. (2020). Statistics for The Behavioral Sciences (10th ed.). Cengage Learning.