Assignment 11: Single Sample T-Test And Factors That Determi

Assignment 11 Single Sample T Testwhat Factor Determines Whether You

What factor determines whether you should use a z-test or a t-test statistic for a hypothesis test? A sample of n = 16 individuals is selected from a population with a mean of μ = 74. After a treatment, the sample variance is s² = 64, and the sample mean is M = 77. If the treatment has a 3-point effect, is this sufficient to conclude a significant treatment effect using a two-tailed test at α = 0.05? If the treatment has a 6-point effect, M = 80, is this sufficient for statistical significance under the same conditions? Additionally, a company notes the previous year's average absences at μ = 5.8; after offering free flu shots to 100 employees, the mean number of absences decreased to M = 3.6 with SS = 396. Does this data indicate a significant decrease? For the spatial skills task with a normative mean μ = 15, testing 7 individuals with brain injuries yields data as follows: 12, 16, 9, 8, 10, 17, 10. Does right-hemisphere damage significantly impair performance at α = 0.05 one-tailed? The assignment also discusses the assumptions for the single sample t-test, hypothesis formulation, critical region determination, and the calculation and interpretation of the t-statistic.

Paper For Above instruction

The selection between a z-test and a t-test in hypothesis testing fundamentally depends on the knowledge of the population standard deviation (σ). Specifically, when the population standard deviation is known, the z-test is appropriate; however, in most practical research scenarios where σ is unknown, the t-test is the suitable choice. This distinction stems from the fact that the t-test accounts for additional variability introduced by estimating the population standard deviation from a sample, especially with small sample sizes, generally n

To understand why the factors determining the choice of test are critical, it’s essential to examine the underlying assumptions and conditions for using each test. The z-test assumes a normal distribution of the population and the availability of the population standard deviation. Conversely, the t-test requires the assumption that the population from which the sample is drawn is approximately normally distributed, especially relevant with small samples, but does not assume known population standard deviation; instead, it relies on the sample standard deviation as an estimate (Leech, Barrett, & Morgan, 2015). When the sample size is large (n ≥ 30), the sampling distribution of the mean approximates normality due to the Central Limit Theorem, allowing the use of a z-test even if σ is unknown, provided the sample standard deviation closely approximates the population SD (Cohen et al., 2018). In general, for smaller samples with unknown population variance, the t-test is more appropriate because it better accounts for sampling variability.

Applying these principles to the examples provided, the first scenario involves a sample of 16 individuals (n = 16) with an unknown population standard deviation. Since the population variance is unknown and the sample size is less than 30, a t-test should be used to assess the significance of the treatment effect. The calculation for the t-statistic hinges on the sample mean, hypothesized population mean, and the estimated standard error. The formula is: t = (M - μ) / (s / √n), where s is the sample standard deviation derived from s² = 64, thus s = 8.

Using this, for the 3-point effect with M = 77:

Standard error = s / √n = 8 / √16 = 8 / 4 = 2

t = (77 - 74) / 2 = 3 / 2 = 1.5

The critical t-value for df = 15 at α = 0.05 (two-tailed) is approximately ±2.131 (from t-distribution tables). Since the calculated t (1.5) does not exceed ±2.131, the result is not statistically significant at this effect level.

In contrast, for the 6-point effect with M = 80:

t = (80 - 74) / 2 = 6 / 2 = 3

Here, the calculated t (3) exceeds 2.131, indicating a statistically significant result, and thus, the 6-point treatment effect can be concluded to be significant under the specified conditions.

In the second scenario, the company assessing the impact of flu shots on employee absences, the sample size is 100, and the sample variance is known indirectly via SS = 396. The mean number of absences is 3.6. The null hypothesis posits no change or increase in absences (μ ≥ 5.8), against the alternative hypothesis that absences have decreased (μ

Calculating the z-statistic:

Standard error = s / √n = 2 / √100 = 2 / 10 = 0.2

Z = (M - μ) / standard error = (3.6 - 5.8) / 0.2 = -2.2 / 0.2 = -11

The critical z-value at α = 0.05 (one-tailed) is approximately -1.645. Since -11

In the third case involving spatial skills of individuals with right-hemisphere damage, the population mean is 15 correct solutions. The sample size is 7, with individual data points. The null hypothesis is that right-hemisphere damage does not impair spatial performance (μ = 15), opposed by the alternative that it does impair performance (μ

Calculating the sample mean and standard deviation:

• Sum of data: 12 + 16 + 9 + 8 + 10 + 17 + 10 = 82
• Mean: 82 / 7 ≈ 11.71

Calculating sample variance involves computing deviations from the mean, squaring, summing, and dividing by n - 1:

Sum of squared deviations = (12 - 11.71)^2 + (16 - 11.71)^2 + (9 - 11.71)^2 + (8 - 11.71)^2 + (10 - 11.71)^2 + (17 - 11.71)^2 + (10 - 11.71)^2 ≈ 35.14

Sample variance s² ≈ 35.14 / 6 ≈ 5.86, and sample standard deviation s ≈ √5.86 ≈ 2.42.

Then, compute the t-statistic:

t = (M - μ) / (s / √n) = (11.71 - 15) / (2.42 / √7) ≈ -3.29 / (2.42 / 2.65) ≈ -3.29 / 0.91 ≈ -3.61

Critical t-value at df = 6 for α = 0.05 (one-tailed) is approximately -1.943. Since -3.61

In summary, the key factor determining whether to use a z-test or a t-test hinges on whether the population standard deviation is known and the sample size, with the t-test being appropriate with unknown population variance and small samples. The examples demonstrate the application of these principles in practical research scenarios, emphasizing the importance of understanding assumptions and conditions for statistical tests.

References

• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Leech, N., Barrett, K. C., & Morgan, G. A. (2015). SPSS for intermediate statistics: Use and interpretation. Routledge.
• Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2018). Applied multiple regression/correlation analysis for the behavioral sciences. Routledge.
• Sullivan, M., & Ballantine, J. (2016). Research methods in education. Pearson.
• Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage Publications.
• Leech, N., Barrett, K. C., & Morgan, G. A. (2015). SPSS for intermediate statistics: Use and interpretation. Routledge.
• Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2018). Applied multiple regression/correlation analysis for the behavioral sciences. Routledge.
• Sullivan, M., & Ballantine, J. (2016). Research methods in education. Pearson.
• Jenkins, S. (2011). Statistical methods for health care research. Jones & Bartlett Learning.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics. Pearson.