Question

A test of abstract reasoning is given to a random sample of students before and after completing a formal logic course. The results are shown below. Before 74 83 75 88 84 63 93 84 91 77 After 73 77 70 77 74 67 95 83 84 75 Suppose a researcher wants to test whether the mean score after the course differs from the mean score before the course. Assume that the mean score differences of Before and After completing a formal logic course follow a normal distribution. Let up = population mean score differences of Before and After completing a formal logic course. (Hint: The data given can be considered as paired data) Question 10 (4 points) State the null and alternative hypothesis. o Ho : Mp > 0 H:Hp < 0 Ho : Mp + 0

Answer 1

(10) The correct option is (3), i.e.,
Houp=0
and
H:HD +0

So, the null and alternative hypothesis are:

Null hypothesis, , i.e., the true population mean score difference of Before and After completing a formal logic course is NOT DIFFERENT.

Houp=0

Alternative hypothesis, , i.e., the true population mean score difference of Before and After completing a formal logic course is SIGNIFICANTLY DIFFERENT.

H:HD +0

_________________________

(11) The correct option is (2), i.e., 2.37

The formula for calculating the test statistic for paired test is-

D - HD t =

and it follows a t-distribution with degrees of freedom, $\mathbf{df=n-1}$

Now before calculating the test-statistic we first need to calculate

$\mathrm{\bar{D}}$: sample mean difference between n matched pairs.

number of matched pairs

$s_{D}:$ sample standard deviation of n difference of matched pairs.

	$\mathbf{Before}$	$\mathbf{After}$	$\mathbf{D=Before-After}$	$\mathbf{D-\bar{D}}$	$\mathbf{(D-\bar{D})^{2}}$
1.	74	73	1	-2.7	7.29
2.	83	77	6	2.3	5.29
3.	75	70	5	1.3	1.69
4.	88	77	11	7.3	53.29
5.	84	74	10	6.3	39.69
6.	63	67	-4	-7.7	59.29
7.	93	95	-2	-5.7	32.49
8.	84	83	1	-2.7	7.29
9.	91	84	7	3.3	10.89
10.	77	75	2	-1.7	2.89
			$\Sigma{D}=37$ $\bar{D}=\frac{\Sigma{D}}{n}=\frac{37}{10}=\mathbf{3.7}$		$\mathbf{\Sigma{(D-\bar{D})^{2}}=220.1}$

$s_{D}=\sqrt{\frac{\Sigma{(D-\bar{D})^{2}}}{n-1}}=\sqrt{\frac{220.1}{10-1}}=\mathbf{4.945255864}$

Calculation for test-statistic:

$\mathbf{t=\frac{\bar{D}-\mu_{D}}{\frac{s_{D}}{\sqrt{n}}}}=\frac{3.7-0}{\frac{4.945255864}{\sqrt{10}}}=2.365990287\approx \mathbf{{\color{Blue} 2.37}}$

So, the test-statistic is calculated as $\mathbf{t=2.37}$

____________________________

(12) The correct option is (4), i.e., $\mathbf{\pm2.2622}$

Since the significance level is given as and we are testing a two-tailed hypothesis, then the critical value is given as-

= 0,05

$\mathrm{t_{(\frac{\alpha}{2},df=n-1)}=t_{(\frac{0.05}{2},df=10-1=9)}=t_{(0.025,df=9)}=\pm2.262157\approx\mathbf{ {\color{Blue} \pm2.2622}}}$

________________________

(13) The correct option is (1), i.e., $\mathbf{Reject\:H_{0}}$

Rejection rule for this hypothesis is: We reject $H_{0}$ if either or .

Since, $t=2.37>2.2622\Rightarrow \mathbf{{\color{Teal} We\:reject\:null\:hypothesis\:H_{0}}}$

At significance level of
= 0,05
sample data provides sufficient evidence to reject null hypothesis $H_{0}$ , hence we reject null hypothesis.

In other words, at based on sample data we can conclude that, "there is a significant mean score difference in score of student Before and After completing a formal logic course."

= 0,05