A school experiments with two teaching approaches for mathematics. In one classroom the material is taught using lectures only. In the other classroom lectures are mixed in with the use of manipulatives, computers, and collaborative activities. Do the data below support the claim that students in the classroom with multiple techniques do better. (a=.05)
Lecturen=20x=76.0s=2.1
Multiplen=32x=88.5s=0.4
To determine if the data supports the claim that students in the classroom with multiple techniques (lectures mixed with manipulatives, computers, and collaborative activities) do better, we can conduct a hypothesis test.
The first step is to state the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: There is no significant difference in the mean scores between the two teaching approaches.
Ha: Students in the classroom with multiple techniques have higher mean scores.
Next, we need to determine the appropriate test statistic. Since we are comparing the means of two independent populations and the sample sizes in each group are relatively small (n<30), we can use a t-test.
Now, let's calculate the test statistic (t-value) using the given data:
For the lecture-only classroom:
Sample size (n1) = 20
Sample mean (x1) = 76.0
Sample standard deviation (s1) = 2.1
For the multiple techniques classroom:
Sample size (n2) = 32
Sample mean (x2) = 88.5
Sample standard deviation (s2) = 0.4
The formula for calculating the pooled standard deviation (Sp) is:
Sp = sqrt(((n1-1)s1^2 + (n2-1)s2^2)/(n1+n2-2))
Substituting the values, we get:
Sp = sqrt(((20-1)*(2.1)^2 + (32-1)*(0.4)^2)/(20+32-2))
Sp = sqrt((39.90 + 12.96)/50)
Sp = sqrt(0.866)
Now we can calculate the t-value using the formula:
t = (x1 - x2) / (Sp * sqrt(1/n1 + 1/n2))
Substituting the values we have:
t = (76.0 - 88.5) / (sqrt(0.866) * sqrt(1/20 + 1/32))
t = -12.5 / (0.294 * 0.312)
t = -12.5 / 0.0916
t ≈ -136.4
To determine if this t-value is statistically significant at the significance level of α = 0.05, we need to compare it to the critical t-value from the t-distribution.
Since the degrees of freedom in this case are (n1-1) + (n2-1) = 19 + 31 = 50, the critical t-value can be found using a t-table or a statistical software.
Comparing the calculated t-value (-136.4) to the critical t-value, we find that it far exceeds the critical t-value at α = 0.05.
Therefore, based on the given data and conducting a t-test, we can conclude that there is strong evidence to support the claim that students in the classroom with multiple techniques have significantly higher mean scores.