Respuesta :
Answer:
1) D) 2
2) A) -9.92 to -2.08
3) D) -3
4) B) 0.0028
5) A) There is a statistically significant difference in the average final examination scores between the two classes
Step-by-step explanation:
The standard deviation s of the difference between the means of the two populations can be calculated as:
[tex]s=\sqrt{\frac{s_{1} ^{2} }{n1}+\frac{s_{2} ^{2} }{n2}}[/tex]
Where [tex]s_1^{2}[/tex] is the variance of the first sample, [tex]s_2^{2}[/tex] is the variance of the second sample, n1 is the size of the first sample and n2 is the size of the second sample. So, The standard deviation of the difference between the means of the two populations is:
[tex]s=\sqrt{\frac{112.5}{45}+\frac{54}{36}}=2[/tex]
A 95% confidence interval for the difference between the two pop. means can be calculated as:
[tex](x1-x2)-z_{\alpha/2} s\leq(m1-m2)\leq (x1-x2)+z_{\alpha/2} s[/tex]
Where (x1-x2) is the difference between the two sample means, 1-α is equal to 95% and (m1-m2) is the difference between the two population means.
Then, replacing, x1 by 82, x2 by 88, s by 2 and [tex]z_{\alpha/2}[/tex] by 1.96, we get:
[tex](82-88)-1.96(2)\leq(m1-m2)\leq (82-88)+1.96(2)[/tex]
-6 - 3.92 ≤ m1-m2 ≤ -6 + 3.92
-9.92 ≤ m1-m2 ≤ -2.08
On the other hand, we can formulate a test hipotesis as:
H0: m1-m2 = 0
H1: m1-m2 ≠ 0
So, the test statistic for the difference between the two pop. means can be calculated as:
[tex]z=\frac{(x1-x2)-(m1-m2)}{s}[/tex]
Replacing the values, we get:
[tex]z=\frac{(82-88)-(0)}{2}=-3[/tex]
Therefore, the p-value for the difference between the two pop. means can be calculated as:
p-value = 2P(Z<-3) = 2(0.0014)=0.0028
Where the probability P(Z<-3) is founded on a z-score table.
Then, if the p-value is less than the level of significance, we can reject the null hypothesis or H0 and said that there is a statistically significant difference in the average final examination scores between the two classes.
