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Proportions play a critical role in hypothesis testing for comparing groups. In this exercise, we look at two groups of grapevines, each treated with a different insecticide. The exercise requires us to calculate the proportion of vines that were infested in each group. This is done to compare the effectiveness of the two treatments. A proportion is essentially a part of the whole, calculated by dividing the part by the whole. Here, we divide the number of infested vines by the total number of vines to find the proportions for each group.

• Pernod 5 Group: The proportion of infested vines is calculated as \( \hat{p}_1 = \frac{88}{400} = 0.22 \).


• Action Group: Similarly, for the Action treated vines, the proportion is \( \hat{p}_2 = \frac{63}{400} = 0.1575 \).

These values tell us that 22% of the vines treated with Pernod 5 were infested compared to 15.75% for those with Action. Calculating these proportions helps establish a basis for comparative analysis, which is foundational in hypothesis testing.

The significance level, denoted as \( \alpha \), is a threshold set to determine the strength of the evidence needed to reject the null hypothesis. For this problem, the significance level is set at 0.05. The choice of 0.05 as the significance level is common in many scientific studies and gives a 5% risk of concluding that a difference exists when there is none (Type I error).

• Setting \( \alpha = 0.05 \) means that if the probability of observing our test statistic is less than 5% assuming the null hypothesis is true, we will reject the null hypothesis.


• In essence, it provides a cut-off point for decision-making, minimizing the chances of wrong inference.

Using this significance level, we compare our calculated test statistic (in this case, a Z-value) with the critical value from the statistical distribution (normal distribution for Z-tests). If our statistic falls beyond the significance threshold, it indicates sufficient evidence to support the alternative hypothesis.

The Z-value, or Z-score, is a measure of how many standard deviations a data point is from the mean of your data set. In hypothesis testing, the Z-value helps to determine whether the result from comparing two proportions is statistically significant. For this exercise, the Z-value is calculated using the formula:\[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \]where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, and SE is the standard error of the difference between two proportions.

• In this instance, the Z-value is computed as \( \frac{0.22 - 0.1575}{0.0289} \approx 2.161 \).


• This value tells us how much our observed difference deviates from what is expected under the null hypothesis of no difference.

For a significance level of 0.05 in a two-tailed test, a critical Z-value is approximately \( \pm 1.96 \). Since the calculated Z-value (2.161) is greater than 1.96, we reject the null hypothesis, suggesting a statistically significant difference between the two groups.