Question

Test scores of ten individuals before and after a training program are shown below.

Individual - Score After Program - Score Before Program
1 - 57 - 59
2 - 62 - 57
3 - 60 - 60
4 - 63 - 66
5 - 69 - 68
6 - 63 - 59
7 - 74 - 72
8 - 56 - 52
9 - 64 - 58
10 - 64 - 63

What is the sum of positive signed ranks (T+)?

The mean µT is?

The value of σT is?

At α = 0.05, what can be concluded about the effectiveness of the training program?

At α = 0.2, what can be concluded about the effectiveness of the training program?

Answer 1

Answer on Question #57326 - Math - Statistics and Probability

Test scores of ten individuals before and after a training program are shown below.

Question

What is the sum of positive signed ranks $(T^{+})$

Solution

$$T ^ {+} = 8 + 1. 5 + 6. 5 + 3. 5 + 6. 5 + 9 + 1. 5 = 3 6. 5$$

Question

The mean $\mu_T$ is?

Solution

Using Large Sample Approximation to Signed-Rank Statistic

$$\mu_ {T} = \frac {n (n + 1)}{4} = \frac {9 (9 + 1)}{4} = 2 2. 5$$

Question

The value of $\sigma_T$ is?

Solution

Using Large Sample Approximation to Signed-Rank Statistic

$$\sigma_ {T} = \sqrt {\frac {n (n + 1) (2 n + 1)}{2 4}} = \sqrt {\frac {9 (9 + 1) (2 \cdot 9 + 1)}{2 4}} = 8. 4 4$$

Question

At $\alpha = 0.05$ , what can be concluded about the effectiveness of the training program?

Solution

Let $H_0$ : No difference before and after a training program

$H_{1}$ : Scores after a training program are better than before

Using Large Sample Approximation to Signed-Rank Statistic

$$z = \frac {T ^ {+} - \mu_ {T}}{\sigma_ {T}} = \frac {36.5 - 22.5}{8.44} = 1.659.$$

The p-value is

$$P(Z > 1.659) = 0.048558.$$

Since p-value is less than alpha, we reject $H_0$ and we can conclude that the training program was effective.

**Question**

At $\alpha = 0.2$, what can be concluded about the effectiveness of the training program?

**Solution**

Reject $H_0$, hence the training program was effective.

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