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Arsenic in Chicken Data 4.5 on page 228 discusses a test to determine if the mean level of arsenic in chicken meat is above 80 ppb. If a restaurant chain finds significant evidence that the mean arsenic level is above \(80,\) the chain will stop using that supplier of chicken meat. The hypotheses are $$ \begin{array}{ll} H_{0}: & \mu=80 \\ H_{a}: & \mu>80 \end{array} $$ where \(\mu\) represents the mean arsenic level in all chicken meat from that supplier. Samples from two different suppliers are analyzed, and the resulting p-values are given: Sample from Supplier A: p-value is 0.0003 Sample from Supplier B: p-value is 0.3500 (a) Interpret each p-value in terms of the probability of the results happening by random chance. (b) Which p-value shows stronger evidence for the alternative hypothesis? What does this mean in terms of arsenic and chickens? (c) Which supplier, \(\mathrm{A}\) or \(\mathrm{B}\), should the chain get chickens from in order to avoid too high a level of arsenic?

Step by step solution

01

Understand the P-value

A p-value is a measure of the probability that an observed difference could have occurred just by random chance. The lower the p-value, the greater the statistical evidence is against the null hypothesis. As a general rule, a p-value less than or equal to 0.05 is usually taken to mean there is strong evidence against the null hypothesis.

02

Interpret the p-values given

Supplier A has a p-value of 0.0003. This means there is a 0.03% chance that the arsenic levels seen occurred by chance alone, assuming the null hypothesis is true (i.e. the true mean is 80 ppb). This is very strong evidence against the null hypothesis. For Supplier B, the p-value of 0.3500 or 35.00% is a much higher probability, indicating that the observed arsenic levels are much more likely to be due to chance, when assuming that the true mean is 80 ppb.

03

Compare the p-values

The p-value from Supplier A is significantly lower than that from Supplier B. Therefore, the data presents stronger evidence against the null hypothesis for Supplier A. This indicates that the mean arsenic level in chickens from Supplier A is more likely to be greater than 80 ppb.

04

Determine the supplier

Given the objective to avoid a high level of arsenic, it would be advisable to choose Supplier B. Despite having a higher p-value, this only indicates that no significant evidence was found to contradict the null hypothesis, i.e., the mean arsenic level from Supplier B is likely to be 80 ppb or less.

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Key Concepts

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

P-value Interpretation

When conducting hypothesis testing, understanding the p-value is crucial. The p-value represents the probability of obtaining test results at least as extreme as the ones observed, under the assumption that the null hypothesis is true. Essentially, it helps us determine how likely our results could have occurred by random chance.

A smaller p-value suggests stronger evidence against the null hypothesis. For example, a p-value of 0.0003 means there's only a 0.03% chance that the observed results occurred by chance, assuming the null hypothesis holds true. Conversely, a larger p-value, such as 0.3500, indicates a higher probability that the results can be attributed to random variation.

Understanding p-values allows you to assess whether the evidence is strong enough to support the alternative hypothesis over the null hypothesis. Remember, the typical threshold to reject the null hypothesis is a p-value of 0.05 or below.

• A low p-value (<0.05) signals strong evidence against the null hypothesis.
• A high p-value (>0.05) suggests the evidence is not strong enough to reject the null hypothesis.

Null Hypothesis

The concept of the null hypothesis is fundamental to hypothesis testing. In statistical terms, the null hypothesis, denoted as \(H_0\), posits that there is no effect or no difference; it represents a default or a skeptical perspective on the subject of study. In the context of arsenic levels in chicken meat, the null hypothesis is \(\mu = 80\text{ ppb}\).

The null hypothesis is what scientists and researchers aim to challenge or disprove with their data. The alternative hypothesis, denoted as \(H_a\), presents the opposing claim, suggesting the mean arsenic level is greater than 80 ppb in this scenario.

The strength of evidence against the null hypothesis is often measured by calculating the p-value. If the evidence against \(H_0\) is strong enough (i.e., the p-value is low), researchers take this as support for the alternative hypothesis.

• Null Hypothesis \(H_0\): Represents no change or no effect.
• Alternative Hypothesis \(H_a\): Suggests a change or effect is present.
• Decision to reject or not reject \(H_0\) is based on p-value and significance level.

Statistical Significance

Statistical significance plays a crucial role in analyzing experiment results. It helps determine whether an observed effect or difference is not due just to chance. To identify statistical significance, researchers compare the p-value to a critical value known as the significance level (often denoted as \(\alpha\)).

Common practice sets this level at 0.05, which means there's a 5% probability that the observed results happened by chance. If the p-value is less than \(\alpha\), the results are deemed statistically significant, suggesting there's enough evidence to reject the null hypothesis in favor of the alternative.

For instance, in this exercise, the p-value for Supplier A was 0.0003, far below the 0.05 threshold. This indicates strong statistical significance, implying the mean arsenic level is more likely above 80 ppb. Meanwhile, for Supplier B, the p-value of 0.3500 doesn't meet the threshold, suggesting no statistical significance.

• Statistical significance is determined by comparing p-value to significance level \(\alpha\).
• A result is statistically significant if p-value < \(\alpha\), typically 0.05.
• Statistical significance provides evidence to support or reject hypotheses.