A mathematics professor is interested in increasing the mathematical ability of

Question

A mathematics professor is interested in increasing the mathematical ability of college students in remedial math classes. The professor obtains a random sample of ten students enrolled in remedial math this semester and begins the semester teaching these ten students using the new "Skills Building Program" (SBP) that the professor devised. The final exam these ten students are given is the same final exam the professor has used for the past 15 semesters, where the mean score is 12 and the standard deviation is 4 across these previous 15 semesters (mu = 12 and sigma = 4). High scores on this final indicate better mathematical performance. Test the professor's hypothesis that the new SBP method improves scores on the final exam (and hence increases mathematical ability). Final exam scores of students with new SBP method: 12, 10, 14, 13, 17, 12, 13, 17, 13, 15 With alpha = 01, complete step 4 of the hypothesis testing procedure, what decision and conclusion should the researcher make? The researcher should retain Ho and conclude the new SBP method improves scores on the final The researcher should retain H_0 and conclude the new SBP method docs not improve scores on the final The researcher should reject H_0 and conclude the new SBP method improves scores on the final The researcher should reject H_0 and conclude the high scores on the SBP are directly related to high scores on the final The researcher should retain H_0 and conclude the new SBP method improves scores on the final

Explanation / Answer

Solution:-

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: 1 - 2 = 0, new SBP method does not improve the scores on the final exam.
Alternative hypothesis: 1 - 2 0, new SBP method does improve the scores on the final exam.

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s12/n1) + (s22/n2)]
SE = sqrt[(42/15) + (2.222/10)] = 1.2488

DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] }
DF = (42/15 + 2.222/10)2 / { [ (42 / 15)2 / (14) ] + [ (2.222 / 10)2 / (9) ] }
DF = 2.43206 / (0.08126984126 + 0.0269879184) = 22.46 or 22

t = [ (x1 - x2) - d ] / SE = [ (12 - 13.6) - 0 ] /1.2488 = -1.28

where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 22 degrees of freedom is more extreme than -1.28; that is, less than -1.28 or greater than 1.28.

We use the t Distribution Calculator to find P(t < -1.28)

The P-Value is 0.213609.
The result is not significant at p < 0.01.

Interpret results. Since the P-value (0.213609) is greater than the significance level (0.01), we cannot reject the null hypothesis.

Conclusion. The researcher should retain H0 and conclude the new SBP method does not improve scores on the final.

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