91Ó°ÊÓ

Proportion comparison is a fundamental part of hypothesis testing when we want to determine if there are differences between two groups. In our garlic example, we are looking at whether ionizing radiation results in a higher proportion of marketable garlic bulbs compared to untreated garlic bulbs. To do this, we start by finding the sample proportions.

For irradiated garlic bulbs, we have a proportion of 153 marketable out of 180, which gives us \( \hat{p}_1 = \frac{153}{180} = 0.85 \). On the other hand, for untreated bulbs, the proportion is 119 out of 180, or \( \hat{p}_2 \approx 0.6611 \). By comparing these proportions, our goal is to see if there is a significant difference suggesting that one is more effective in preserving the garlic bulbs than the other.

In hypothesis testing, this section is crucial because calculating these proportions helps set up the basis for testing our hypotheses. We are particularly interested in knowing whether the observed difference between these two proportions is statistically meaningful or just due to random chance. A meaningful difference would support the claim that irradiation significantly increases marketability.

Calculating the standard error is a vital step in hypothesis testing, especially when comparing proportions from two samples. The standard error gives us an idea of how much variance we can expect in the sample means or proportions, acting as a measure of the reliability of the sample.

In the garlic study, we first calculate an overall sample proportion:\[ \hat{p} = \frac{153 + 119}{360} \approx 0.7556 \]. This value serves as a pooled sample proportion which helps in adjusting our standard error calculation.

We then calculate the standard error (SE) for the difference between the two proportions:\[ SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.7556(1-0.7556)\left(\frac{1}{180} + \frac{1}{180}\right)} \approx 0.0493 \].

This calculated standard error indicates how the difference in the sample proportions might fluctuate from sample to sample, provided the null hypothesis is true. Essentially, it helps us understand if our observed difference is substantial enough, considering the variation we expect.

The Z-test statistic is a key component in making decisions during hypothesis testing. It quantifies the difference between the observed sample data and the null hypothesis. In this case, it helps us verify if the difference in the proportion of marketable garlic bulbs between the treated and untreated groups is statistically significant.

To find the Z-test statistic, we use the formula:\[ z = \frac{(\hat{p}_1 - \hat{p}_2)}{SE} = \frac{(0.85 - 0.6611)}{0.0493} \approx 3.827 \].

The numerator of this formula represents the observed difference in sample proportions, while the denominator, the standard error, accounts for the expected variability due to random sampling.

In our case, the calculated Z-value is approximately 3.827. This value is compared against a critical Z-value, typically from the standard normal distribution, to determine statistical significance. Because 3.827 is greater than the critical value for a one-tailed test at a 5% significance level, which is 1.645, we reject the null hypothesis. This indicates that there is significant evidence to suggest that irradiation improves the marketability of garlic bulbs.