Question

3. There are seven positive differences in the paired observations, five negative differences and 4 zero differences.
3.1 If a sign test is to be conducted: What is the value of n?
3.2 If a sign test is to be conducted: What is the value of the test statistic?
3.3 With α=0.05 in a two – tailed test, will the null hypothesis accepted or rejected?

Answer 1

In a sign test, we analyze paired observations to determine whether there is a significant difference between them. Let's break down your questions step by step:

### 3.1 Value of \( n \)

In a sign test, \( n \) represents the number of non-zero differences. To find \( n \), we sum the counts of positive and negative differences.

Given:
- Positive differences = 7
- Negative differences = 5
- Zero differences = 4 (these are not included in \( n \))

So,
\[
n = \text{Positive differences} + \text{Negative differences} = 7 + 5 = 12
\]

### 3.2 Value of the test statistic

In a sign test, the test statistic is the smaller of the counts of positive and negative differences.

From the information provided:
- Positive differences = 7
- Negative differences = 5

The test statistic \( T \) is given by:
\[
T = \min(\text{Positive differences}, \text{Negative differences}) = \min(7, 5) = 5
\]

### 3.3 Significance of the test statistic with \( \alpha = 0.05 \)

To determine whether to accept or reject the null hypothesis, you would need to compare your test statistic to the critical value from the binomial distribution. In a sign test, the null hypothesis typically states that there is no difference between the pairs.

Given:
- \( n = 12 \)
- \( \alpha = 0.05 \)

This is a two-tailed test, meaning we are looking for critical values in both tails of the distribution. For a two-tailed test with \( n = 12 \):

1. Total possible outcomes of the signs can be modeled as a binomial distribution \( B(n, p) \) where \( p = 0.5 \).
2. You need to find the critical values for \( T \) using the binomial distribution.

Using a binomial distribution:
- The critical values for \( n = 12 \) and \( \alpha = 0.05 \) (for a two-tailed test) would generally involve finding the lower and upper \( 2.5\% \) tails.

Calculating the critical values using binomial tables or software, you would find:
- Critical value for \( T \) = 2 (lower) and 10 (upper).

Since the test statistic \( T = 5 \), which lies between 2 and 10:

### Conclusion:

Since \( T = 5 \) is not in the critical region defined by \( T \leq 2 \) or \( T \geq 10 \):
- **The null hypothesis is not rejected.**

In summary:

1. \( n = 12 \)
2. Test statistic \( T = 5 \)
3. Null hypothesis not rejected (accept the null hypothesis).