Question

TFA programs attempt to address the shortage of qualified teachers by placing uncertified instructors in schools with acute needs. A 1999–2000 study compared students taught by certified teachers vs under certified teachers in the same schools. Reading scores of students of certified teachers averaged 35.62 points with standard deviation 9.31. Those instructed by under certified teachers had mean 32.48 points with standard deviation 9.43 points on the same test. Each group had 44 students. Is there evidence of lower scores with uncertified teachers?

la) Write the hypotheses statements below to test the rancher's claim: Ho : Ha : lb) Which hypothesis represents lower scores of the uncertified? Circle one: Null Hypothesis (Ho) or Alternative Hypothesis (Ha) 2) Explain what type of hypothesis testing you will perform and whether conditions are met. 3a) Test this hypothesis using a significance level of a = 5%. (SHOW WORK!) Clearly label a sketch with appropriate shading and calculate the test statistic (show formula and work). 3b) Would you reject or fail to reject the null hypothesis? Circle one: Reject Ho or Fail to Reject Ho Explain your choice: 4a) Using a significance level of a = 5%, write a conclusion in the context of this problem:

Answer 1

Answers:  1) a) To determine if there Is any evidence of lower scores with uncertified teachers we construct our null and alternative hypotheses as H0: mu1 = mu2 vs Ha: mu1 > mu2 where mu1 and mu2 are the unknown means of the scores of students of certified and uncertified teachers respectively.

1) b)Here the alternative hypothesis  represents lower scores of the uncertified.

2) We are using an one-sided test here. The assumption here is that both the samples are drawn from normal populations. Since sample sizes = 44 is quite large, by CLT we can say that the required assumption is met.

3) a) The test statistic used for this test is T= (x1bar-x2bar)/sqrt((s1*s1/n1)+(s2*s2/n2)) where x1bar, x2bar are the sample means, s1, s2 are the sample standard deviations and n1,n2 are the sample sizes. sqrt refers to the square root function.

We reject H0 if T(observed) > t(alpha,v) where t(alpha,v) is the upper alpha point of a Student's t distribution with "v" degrees of freedom. Alpha is the level of significance. v = (((s1/n1)+(s2/n2))^2) / (((s1/n1)^2)/(n1-1))+(((s2/n2)^2)/(n2-1))

3) b) Here T(observed) = 1.571783 and t(alpha,v) =  1.662768. Thus we see that T(observed) > t(alpha,v). Thus we fail to reject H0 .

4)a) We conclude on the basis of the given sample at a 5% level of significance that there is insufficient evidence to support the claim that there is presence of lower scores with uncertified teachers.