Question

The University of the Fort Hare wishes to investigate if a new variety of wheat differs in resistant to diseases compared to an old variety. It is known that this disease strikes approximately 15% of all plants of the variety. A field experiment was conducted, and of 120 new plants, 20 become infected. Test at 1% level of significance if indeed the new variety of wheat differs in resistant to diseases compared to an old variety. Make you decision.

Answer 1

To determine if the new variety of wheat differs in resistance to diseases compared to the old variety, we will conduct a hypothesis test for proportions.

### Step 1: Define the Hypotheses

- Null Hypothesis (\(H_0\)): The proportion of infections in the new variety is equal to the proportion of infections in the old variety. That is, \(p = 0.15\).
- Alternative Hypothesis (\(H_a\)): The proportion of infections in the new variety is different from the old variety. That is, \(p \neq 0.15\).

### Step 2: Gather Sample Data

From the field experiment:
- Sample size (\(n\)) = 120
- Number of infected plants (\(X\)) = 20
- Sample proportion (\(\hat{p}\)) = \(\frac{X}{n} = \frac{20}{120} = \frac{1}{6} \approx 0.1667\)

### Step 3: Determine the Test Statistic

We will use a Z-test for proportion. The test statistic is calculated using the formula:

\[
Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

where:
- \(\hat{p} = 0.1667\) (sample proportion)
- \(p_0 = 0.15\) (population proportion under null hypothesis)
- \(n = 120\) (sample size)

Calculating the standard error:

\[
\text{Standard Error (SE)} = \sqrt{\frac{0.15(1 - 0.15)}{120}} = \sqrt{\frac{0.15 \times 0.85}{120}} = \sqrt{\frac{0.1275}{120}} \approx 0.0327
\]

Now we can compute the test statistic \(Z\):

\[
Z = \frac{0.1667 - 0.15}{0.0327} \approx \frac{0.0167}{0.0327} \approx 0.510
\]

### Step 4: Determine the Critical Value

Since we are conducting a two-tailed test at a significance level (\(\alpha\)) of 0.01, we need to find the critical Z-values for \(\alpha/2 = 0.005\).

Using the standard normal distribution table or calculator, the critical Z-values are approximately \(-2.576\) and \(2.576\).

### Step 5: Make the Decision

Now, we compare our calculated Z value (0.510) with the critical values:

- If \(Z < -2.576\) or \(Z > 2.576\), we reject the null hypothesis.
- If \(-2.576 < Z < 2.576\), we fail to reject the null hypothesis.

In this case, \(0.510\) falls between \(-2.576\) and \(2.576\).

### Conclusion

Since we fail to reject the null hypothesis, we do not have sufficient evidence to conclude that the new variety of wheat differs in resistance to diseases compared to the old variety at the 1% level of significance.