An educational psychologist investigated whether a new spaced-repetition vocabulary app increases the number of words recalled in a foreign language test. Table 1 shows the vocabulary test scores (out of 20) of 11 students measured before and after using the app for a month.

Table 1: Vocabulary test scores (out of 20) before and after using the vocabulary app

Student	Score before	Score after
Student A	12	15
Student B	9	14
Student C	15	18
Student D	11	16
Student E	14	12
Student F	8	13
Student G	10	10
Student H	13	17
Student I	7	11
Student J	11	15
Student K	16	19

When using a Wilcoxon Signed-Ranks Test to analyze the difference, the calculated value of $T\,$ was found to be 5 ( $T = 5$ ).

Table 2: Critical values of $T\,$ for the Wilcoxon Signed-Ranks Test

$N$	$\le 0.05$ (one-tailed)	$\le 0.05$ (two-tailed)
9	8	5
10	10	8
11	13	11

Using the data in Table 1 and Table 2, explain whether the results are significant at $\le 0.05\,$ for a directional (one-tailed) hypothesis, and if the researcher's directional hypothesis can be accepted.

Table 1: Vocabulary test scores (out of 20) before and after using the vocabulary app

Student	Score before	Score after
Student A	12	15
Student B	9	14
Student C	15	18
Student D	11	16
Student E	14	12
Student F	8	13
Student G	10	10
Student H	13	17
Student I	7	11
Student J	11	15
Student K	16	19

When using a Wilcoxon Signed-Ranks Test to analyze the difference, the calculated value of $T\,$ was found to be 5 ( $T = 5$ ).

Table 2: Critical values of $T\,$ for the Wilcoxon Signed-Ranks Test

$N$	$\le 0.05$ (one-tailed)	$\le 0.05$ (two-tailed)
9	8	5
10	10	8
11	13	11

Methods

Question 139

Table 1: Vocabulary test scores (out of 20) before and after using the vocabulary app

Table 2: Critical values of $T\,$ for the Wilcoxon Signed-Ranks Test

Markscheme

Methods

Question 139

Table 1: Vocabulary test scores (out of 20) before and after using the vocabulary app

Table 2: Critical values of $T\,$ for the Wilcoxon Signed-Ranks Test

Markscheme

Methods

Question 139

Table 1: Vocabulary test scores (out of 20) before and after using the vocabulary app

Table 2: Critical values of T T\,T for the Wilcoxon Signed-Ranks Test

Methods Questions

Methods

Question 139

Table 1: Vocabulary test scores (out of 20) before and after using the vocabulary app

Table 2: Critical values of T T\,T for the Wilcoxon Signed-Ranks Test

Methods Questions

Table 2: Critical values of $T\,$ for the Wilcoxon Signed-Ranks Test

Table 2: Critical values of $T\,$ for the Wilcoxon Signed-Ranks Test