The University of the Free State 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. Find the critical value.

Question

GPT-4o mini · Answer

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

Step 1: Set Up the Hypotheses

Null hypothesis (\(H_0\)): The proportion of infected plants in the new variety is equal to the proportion in the old variety, \(p = 0.15\).
Alternative hypothesis (\(H_a\)): The proportion of infected plants in the new variety is not equal to the proportion in the old variety, \(p \neq 0.15\).

Step 2: Collect the Data

In this case:

Sample size (\(n\)) = 120 (the number of new plants)
Number of infected plants in the new variety (\(x\)) = 20
Sample proportion (\(\hat{p}\)) = \(x/n = 20/120 = 1/6 \approx 0.1667\)

Step 3: Calculate the Test Statistic

Calculate the standard error (SE) of the sample proportion: \[ SE = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.15(1 - 0.15)}{120}} = \sqrt{\frac{0.15 \times 0.85}{120}} = \sqrt{\frac{0.1275}{120}} \approx \sqrt{0.0010625} \approx 0.0326 \]
Calculate the z-test statistic: \[ z = \frac{\hat{p} - p}{SE} = \frac{0.1667 - 0.15}{0.0326} \approx \frac{0.0167}{0.0326} \approx 0.512 \]

Step 4: Determine the Critical Value

Since we are performing a two-tailed test at a significance level of \( \alpha = 0.01\), we look for critical values for \( z \) using a standard normal distribution.

The critical values for \( \alpha = 0.01 \) (two-tailed) correspond to:

\( z_{0.005} \) and \( -z_{0.005} \), since \( \alpha/2 = 0.005 \).

The critical values are approximately: \[ z_{0.005} \approx 2.576 \quad \text{and} \quad -z_{0.005} \approx -2.576 \]

Step 5: Make a Decision

Compare the calculated test statistic \( z \) with the critical values:

\( -2.576 < 0.512 < 2.576 \)

Since the test statistic does not fall in the critical region, we fail to reject the null hypothesis.

Conclusion

At the 1% significance level, there is not enough evidence to suggest that the new variety of wheat differs in resistance to diseases compared to the old variety. The critical value for this test is approximately \( \pm 2.576 \).