91Ó°ÊÓ

Sweet Potato Whitefly Suppose that \(10 \%\) of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected, and 25 are found to be infested with whitefly. a. Assuming that the experiment satisfies the conditions of the binomial experiment, do the data indicate that the proportion of infested fields is greater than expected? Use the \(p\) -value approach, and test using a \(5 \%\) significance level b. If the proportion of infested fields is found to be significantly greater than .10 , why is this of practical significance to the agronomist? What practical conclusions might she draw from the results?

Step by step solution

01

Identify the null and alternative hypotheses

The null hypothesis (\(H_0\)) states that the proportion of infested fields is equal to 10%, and the alternative hypothesis (\(H_a\)) states that the proportion of infested fields is greater than 10%. \(H_0: p = 0.1\) \(H_a: p > 0.1\)

02

Calculate the test statistic

We will use the z-test statistic for a proportion, which is given by the formula: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\) where \(\hat{p}\) is the sample proportion, \(p_0\) is the null hypothesis proportion, and \(n\) is the sample size. In this case, we have: \(\hat{p} = \frac{25}{100} = 0.25\) \(p_0 = 0.1\) \(n = 100\) Calculate the z-test statistic: \(z = \frac{0.25 - 0.1}{\sqrt{\frac{0.1(1-0.1)}{100}}}= 4.47\)

03

Calculate the p-value

The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. Using a standard normal distribution table or a calculator, we can find the p-value corresponding to a z-test statistic of 4.47: \(p = P(Z > 4.47) \approx 0.000004\)

04

Compare p-value to the significance level and draw conclusions

Since the p-value (0.000004) is less than the significance level (0.05), we reject the null hypothesis. This indicates that the proportion of infested fields is significantly greater than 10%.

05

Explain the practical significance

If the proportion of infested fields is found to be significantly greater than 10%, this is of practical importance to the agronomist because it indicates that the infestation problem is worse than initially thought. This may lead the agronomist to take more aggressive measures to control the spread of whitefly, or to study the factors that led to the higher infestation rate in order to better manage future cases. In conclusion, the agronomist can use this information to make informed decisions about the management of whitefly infestations in the agricultural area.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In statistical hypothesis testing, the null hypothesis is a foundational concept that acts as a default or starting assumption. In many experiments, including the one described here, it represents the idea that there is no significant effect or difference. In our sweet potato whitefly scenario, the null hypothesis (\(H_0\)) asserts that the proportion of infested fields is exactly 10%.Null Hypothesis (\(H_0\)) Highlights:

• Think of it as "no change" or "nothing unusual happening."
• In this context, it suggests that any observed infestation proportion is just random variation around the expected 10%.
• Provides a baseline for comparing actual data.

Learning to construct a null hypothesis is critical because it grounds scientific testing, helping researchers determine whether observed data is significant or just due to chance.

Alternative Hypothesis

Unlike the null hypothesis, the alternative hypothesis proposes that there is some effect or difference. It is what researchers suspect might be true instead of the null hypothesis. In the whitefly example, the alternative hypothesis (\(H_a\)) suggests that more than 10% of the fields are infested with whiteflies.Key Features of the Alternative Hypothesis (\(H_a\)):

• It challenges the status quo represented by the null hypothesis.
• In this case, it indicates that we suspect a larger problem than initially thought.
• Testing the alternative hypothesis involves looking for evidence that strongly supports this claim over the null hypothesis.

When evidence supports the alternative hypothesis, it can lead to new insights or actions in research, like formulating strategies to address increased infestation levels.

Significance Level

The significance level is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It's denoted by alpha (\(\alpha\)), and in this example, it is set at 5% (0.05). This level represents the probability of rejecting the null hypothesis when it is actually true, which is known as the Type I error.Understanding Significance Level:

• It's essentially a cut-off point. If the calculated p-value is less than \(\alpha\), the null hypothesis is rejected.
• Common choices for significance level are 0.05, 0.01, or 0.10.
• Choosing a lower significance level means you're being more stringent about accepting potential findings.

In practical terms, setting the right significance level helps strike a balance between discovering true effects and minimizing false claims of effects.

p-value

The p-value in a hypothesis test tells us the probability of obtaining results as extreme as the observed results, assuming that the null hypothesis is true. A small p-value indicates that such an extreme observed outcome would be very unlikely under the null hypothesis and suggests that the null hypothesis might be false. Important Points About p-value:

• A smaller p-value points towards evidence against the null hypothesis.
• In the whitefly example, the p-value of 0.000004 is much lower than the significance level of 0.05, hence, the null hypothesis is rejected.
• The p-value does not give the probability that the null hypothesis is true or false. It only helps to decide whether the observed data align nicely with the null hypothesis.

Understanding p-values is crucial. They provide a standardized way to express findings from statistical tests and guide decision-making in scientific investigations.