What Test Would You Use To Determine Whether?

Problem1example What Test Would You Use To Determine Whether The Sta

Determine the appropriate statistical test for the following scenarios:

1. To assess whether the starting salaries for statisticians are greater than $80,000, identify the suitable hypothesis test, specify the null and alternative hypotheses, and explain the procedure.

2. To evaluate whether students in Stat 3011 who have taken AP Statistics perform better than those who have not, determine the appropriate testing procedure.

3. To compare the performance of students on their second exam to their first exam within the same course, specify the statistical test that should be used.

Paper For Above instruction

The selection of appropriate statistical tests is essential for accurately analyzing data and drawing meaningful conclusions in various research scenarios. For the question regarding whether starting salaries for statisticians exceed $80,000, a one-sample t-test for the population mean is appropriate. This test compares the sample mean salary against the hypothesized population mean ($80,000). The null hypothesis (H₀) states that the mean salary equals $80,000, while the alternative hypothesis (H₁) posits that the mean salary is greater than $80,000. The procedure involves calculating the t-statistic based on the sample data, determining the degrees of freedom, and comparing the calculated t-value to the critical t-value from the t-distribution to decide whether to reject H₀ (Gelman & Hill, 2007).

For comparing the performance of students who have taken AP Statistics against those who have not, an independent samples t-test is appropriate. This test evaluates whether the difference in mean exam scores between the two independent groups is statistically significant. The null hypothesis states that there is no difference in mean scores, whereas the alternative suggests a difference exists. The test involves computing the t-statistic based on the sample means, variances, and sizes of the two groups, then referencing the t-distribution for significance testing (McNeil & Frey, 2014).

Regarding the comparison of students' performance on the second versus the first exam within the same course, a paired t-test is suitable. Because the same students take both exams, their scores are paired observations. The null hypothesis posits no difference in mean scores between the two exams, while the alternative hypothesis indicates a difference. The process involves calculating the differences in scores for each student, computing the mean and standard deviation of these differences, and then calculating the t-statistic to examine whether the mean difference is significantly different from zero (Gibbons et al., 2011).

In the second scenario involving teeth whitening, the goal is to test whether the proportion of students who whiten their teeth differs between males and females. The appropriate test is a two-proportion z-test, which compares the two sample proportions to assess if they differ significantly. Under the null hypothesis of no difference, the combined proportion is used to calculate the standard error, and the z-statistic is derived accordingly (Agresti & Finlay, 2009).

a. If the null hypothesis is true, the distribution of the test statistic for the two-proportion z-test is a standard normal distribution (z-distribution), because the z-test relies on the Central Limit Theorem for large samples.

b. Applying the hypothesis testing steps at a significance level (α) of 0.05 involves:

1. Formulating the null hypothesis H₀: p₁ = p₂, where p₁ and p₂ are the true proportions of females and males who whiten their teeth, respectively.
2. Alternative hypothesis H₁: p₁ ≠ p₂, indicating a two-tailed test.
3. Calculating the test statistic z using the formula for two proportions, incorporating the sample proportions and sizes.
4. Determining the critical z-value(s) at α=0.05, which are ±1.96 for a two-tailed test.
5. Comparing the calculated z-value to the critical values to decide whether to reject H₀.

If the z-statistic exceeds the critical value in magnitude, we reject the null hypothesis, suggesting a significant difference in whitening proportions between genders. Otherwise, we fail to reject H₀, indicating insufficient evidence to support a difference.

References

• Agresti, A., & Finlay, B. (2009). Statistical methods for the social sciences (4th ed.). Pearson Education.
• Gibbons, J. D., Hedger, R. C., & Testa, S. M. (2011). Nonparametric statistical inference (4th ed.). Marcel Dekker.
• Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.
• McNeil, C., & Frey, B. (2014). Statistics for Business and Economics (4th ed.). Pearson.