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. Calculate the test statistic.

Answer 1

To investigate the difference in disease resistance between the new variety of wheat and the old variety, a hypothesis test can be conducted. We will use the z-test for proportions.

Hypotheses

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

Given Data

• \( n = 120 \) (sample size)
• \( x = 20 \) (number of infected plants in the new variety)
• Sample proportion (\( \hat{p} \)): \[ \hat{p} = \frac{x}{n} = \frac{20}{120} = \frac{1}{6} \approx 0.1667 \]
• Expected proportion from old variety (\( p_0 = 0.15 \))

Test Statistic

The test statistic for a proportion is calculated using the following formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Step 1: Calculate the standard error (SE)

First, calculate the standard error: \[ \text{SE} = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.15(1 - 0.15)}{120}} = \sqrt{\frac{0.15 \times 0.85}{120}} = \sqrt{\frac{0.1275}{120}} = \sqrt{0.0010625} \approx 0.0326 \]

Step 2: Calculate the z-value

Now, substitute \(\hat{p}\), \(p_0\), and SE into the z-formula: \[ z = \frac{0.1667 - 0.15}{0.0326} \approx \frac{0.0167}{0.0326} \approx 0.512 \]

Step 3: Determine significance level and critical values

With a significance level (\( \alpha = 0.01 \)) for a two-tailed test, we need to find the critical z-values. The critical z-values for a 1% significance level in a two-tailed test are approximately \( \pm 2.576 \).

Step 4: Conclusion

Now we compare the calculated z-value to the critical z-values:

• Calculated \( z \approx 0.512 \)
• Critical values \( z \approx \pm 2.576 \)

Since \( 0.512 \) is within the range of \( -2.576 \) and \( +2.576 \), we do not reject the null hypothesis.

Final Conclusion

At the 1% significance level, there is insufficient evidence to conclude that the new variety of wheat differs in resistance to diseases compared to the old variety.