Week 3 Assignment: Hypothesis Testing And Making Inferences
Week 3 Assignmentapplication Hypothesis Testing Making Inferences Fr
Week 3 Assignment application: Hypothesis Testing: Making Inferences from a Sample 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. Once the hypotheses are stated, researchers move on to data collection. However, once the results come in, the real challenge is to determine if they have meaning; that is, are the results statistically significant? For example, a researcher asks whether the drug for severe depression from this week’s Discussion leads to a different life expectancy than the general population of people with severe depression.
After stating the hypotheses and collecting data, the researcher sees that the mean life expectancy for the people taking the drug is not exactly the same as the rest of the population, but is the difference meaningful (that is, statistically significant) or just due to random variation? There must be a statistically significant difference in order to say that the null hypothesis should be rejected and the people taking the drug really do have a different life expectancy than the population.
Scenario: A researcher asks if eighth-grade students attending a private middle school have higher or lower scores on a test of reading comprehension when compared to the population of eighth-graders attending publicly-funded schools.
A sample of 144 private school eighth-graders take the same exam that all public school 8th graders take at the end of the school year. The private school students have a mean test score of 220.8 and the mean score for the public school students is 204.2, with a standard deviation of 11.4.
Paper For Above instruction
The study under consideration examines whether private middle school eighth-graders outperform their public school counterparts on a reading comprehension test. The research employs a quantitative approach, whereby a sample of private school students' test scores is compared to known population parameters from public school students, in order to determine if there is a statistically significant difference between the two groups. This analysis involves formulating hypotheses, selecting appropriate statistical tests, calculating the test statistic, and interpreting the results within the context of the research question.
Identification of Variables
The independent variable (IV) in this study is the type of school attended—private versus public—since this is the categorical factor that potentially influences reading scores. The dependent variable (DV) is the score on the reading comprehension test, as this is the measurable outcome potentially affected by the type of schooling. The IV is manipulated or grouped based on school type, and its effect on the DV, the test score, is being assessed. Understanding which is which is crucial: the independent variable is what the researcher manipulates or compares, while the dependent variable is what is measured to see if it varies accordingly.
Choice of Statistical Test
Given that the researcher compares the mean test score of private school students to a known population mean from public school students, and considering the sample size (n=144) which is sufficiently large, a z-test for comparing a single sample mean to a population mean is appropriate. The test assesses whether the observed difference in means is statistically significant. Since the research aims to determine whether the private school students' scores are higher or lower than the public school scores without specifying a direction beforehand, a two-tailed test is suitable. A two-tailed test evaluates for deviations in either direction, which aligns with the research question of whether private school students perform better or worse.
Null and Alternative Hypotheses
The null hypothesis (H0) states that there is no difference between the mean reading scores of private school eighth-graders and the population of public school eighth-graders. In other words, attending a private school does not influence reading comprehension scores compared to public schools. Conversely, the alternative hypothesis (HA) posits that there is a difference—private school students have either higher or lower scores than public school students. This non-directional alternative aligns with a two-tailed test, testing for any significant difference regardless of direction.
Calculation of the z Score
Using the provided data, the z-score is calculated with the formula:
z = (Msample - μpopulation) / (σ / √n)
Where:
- Msample
= 220.8 (mean score of private school students)
= 204.2 (mean score of public school students)
Plugging in these values:
z = (220.8 - 204.2) / (11.4 / √144)
√144 = 12, so:
z = 16.6 / (11.4 / 12) = 16.6 / 0.95 ≈ 17.47
Decision Regarding Null Hypothesis
With an alpha level of 0.05, the critical z-value for a two-tailed test is ±1.96. The calculated z-value of approximately 17.47 far exceeds 1.96 in absolute value. Therefore, we reject the null hypothesis, indicating that the difference in mean scores is statistically significant. This substantial z-score suggests that private school students' reading scores are significantly different from those of public school students, with their mean significantly higher than the public school population.
Significance of the Results
The results are statistically significant because the computed z-value exceeds the critical value at the 0.05 significance level. This implies that the observed difference is unlikely to be due to random chance alone and provides evidence against the null hypothesis. In other words, it's highly probable that attending a private school is associated with higher reading comprehension scores among eighth-graders, at least within this sample.
Researcher’s Conclusion
Based on the statistically significant difference, the researcher can conclude that private middle school eighth-graders perform better on reading comprehension tests compared to their public school peers. This suggests an association between the type of school attended and reading achievement, although causality cannot be definitively established from this observational comparison alone. Factors such as curriculum differences, socioeconomic status, and educational resources may contribute to this performance disparity and should be further investigated.
Relationship between z-Scores and Standard Deviation
Z-scores serve as standardized scores indicating how many standard deviations a particular data point is from the mean of a distribution. The calculation of a z-score involves subtracting the population mean from the sample mean and dividing by the standard error (which accounts for the standard deviation and sample size). A larger absolute z-score indicates a greater deviation from the mean, implying more significant differences between groups. It provides a way to compare scores across different distributions and is foundational in hypothesis testing because it standardizes different data points into a common scale, making it easier to assess statistical significance.
References
- Heiman, G. (2015). Behavioral sciences STAT 2 (2nd ed.). Cengage Learning.
- Laureate Education. (2013). Introduction to hypothesis testing [Video].
- StatisticsLectures.com. (2012). Z-scores.
- Khan Academy. (2013). Introduction to normal distribution.
- Texas A & M University. (n.d.). Psychic test.
- Heiman, G. (2015). Describing Data with z-Scores and the Normal Curve. Cengage.
- Heiman, G. (2015). Using Probability to Make Decisions about Data. Cengage.
- StatisticsLectures.com. (2012). Type I and II errors.
- Heiman, G. (2015). Overview of Statistical Hypothesis Testing: The z-Test. Cengage.
- Laureate Education. (2013). Introduction to hypothesis testing.