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A new approach for teaching mathematics was introduced that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a​ three-year period, researchers compared the performances the new approach students with those taught using a traditional curriculum. During the​ study, students in both classes that used the new approach and traditional classes took a Algebra test that did not allow them to use calculators. The accompanying table shows the results. Are the mean scores of the two groups significantly​ different? Address the parts that follow. 

The null hypothesis is the hypothesis to be​ tested, and the alternative hypothesis is what is considered plausible if the null hypothesis is rejected.​ Therefore, the null hypothesis

Upper H 0 : mu Subscript Upper N Baseline minus mu Subscript Upper T Baseline equals 0

and the alternative hypothesis is

Upper H Subscript Upper A Baseline : mu Subscript Upper N Baseline minus mu Subscript Upper T Baseline not equals 0

because the researchers want to know if there is a difference between the mean score for each sample. Since the researchers do not specify whether they are interested in a positive or negative difference between the​ samples, the alternative hypothesis relfects the​ two-tailed nature of the test.

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The​ P-value is the probability of observing a value for a test statistic at least as far from the hypothesized value as the statistic value actually observed if the null hypothesis is true. In this​ case, if the means for the two classes are actually​ equal, there is less than a 1 in​ 10,000 chance of seeing a difference as large as or larger than the observed difference just from natural sampling variation.

ince the stated​ P-value is less than​ 0.0001, which is less than any conventional significance​ level, the researchers can reject the null hypothesis that the mean score from each class is equal. Since the test statistic was found to be

minus

​6.451,

it can be stated that on​ average, students who learn in the new class method do significantly worse on Algebra tests that do not allow them to use calculators than students who learn by traditional methods.

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