Question

Data 4.3 on page 265 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, \(A\) or \(B\), should the chain get chickens from in order to avoid too high a level of arsenic?

Short answer

Based on the p-values, Supplier B should be chosen since there is not strong enough evidence to suggest that its meat contains arsenic levels above 80 ppb. The p-value of 0.3500 implies a 35% likelihood of seeing this data by chance assuming the null hypothesis (mean arsenic level of 80 ppb or less) is true, while the p-value of 0.0003 for Supplier A suggests a 0.03% likelihood, providing strong evidence that the mean arsenic level in their chickens is above 80 ppb.

Step-by-step solution

Understand Hypothesis and The Role of P-value

The hypotheses developed are on the mean arsenic level in the chicken meat from each supplier: - Under the null hypothesis \(H_{0}:\mu=80\), it is being assumed that the mean level of arsenic is 80 ppb. - The alternative hypothesis \(H_{a}:\mu>80\) indicates that the mean arsenic level could be greater than 80 ppb.A p-value represents the probability of obtaining a test statistic as extreme as, or more so, than the one calculated, assuming that the null hypothesis is true. The lesser the p-value, the stronger the evidence is to reject the null hypothesis.

Analysis of p-value of Supplier A

The p-value for Supplier A is 0.0003. This is less than the general threshold for significance of 0.05. This represents a very small probability (0.03%) of getting this data if the null hypothesis is correct. Thus, there is strong evidence to reject the null hypothesis. This suggests that the mean arsenic level in the chicken meat of Supplier A is likely to be above 80 ppb.

Analysis of p-value of Supplier B

The p-value for Supplier B is 0.3500. This is bigger than 0.05, indicating a relatively large probability (35%) of getting this data if the null hypothesis is true. Therefore, there is insufficient evidence to reject the null hypothesis. This means that it is likely the mean arsenic level in Supplier B’s chicken meat is not much above 80 ppb, if at all.

Deciding Between Suppliers

Based on the p-values, Supplier A’s chicken has strong evidence of having an arsenic level above 80 ppb. On the other hand, Supplier B’s chicken does not provide strong enough evidence to suggest that its arsenic level is greater than 80 ppb. Given that the restaurant chain wants to avoid too high a level of arsenic, it would be safer and wiser to choose Supplier B.

Interpreting the Meaning of P-value in Context

It's essential to understand that the p-values obtained don't give direct information about the actual arsenic levels in the chicken. They only provide evidence about how likely it is that the mean arsenic level would exceed 80 ppb, assuming the null hypothesis is true. In the case of Supplier A, it's highly unlikely that the obtained test results would occur by random chance; thus, it’s probable that the arsenic level is above 80 ppb. For Supplier B, the higher p-value signifies that the data obtained could quite possibly happen by random chance even if the null hypothesis were true, therefore, suggesting that the arsenic level is unlikely to exceed 80 ppb.

Key concepts

Null Hypothesis

Understanding the null hypothesis is crucial in the context of hypothesis testing. It's a statement used as a default assumption that there is no effect or no difference—in this case, it posits that the mean arsenic level in chicken meat from a supplier is exactly 80 parts per billion (ppb). The null hypothesis, symbolized as H₀: μ=80, serves as a starting point for statistical analysis. During hypothesis testing, evidence from the data is used to determine whether this assumption can be rejected. If actual observations are significantly different from what we would expect under the null hypothesis, it may lead to rejecting it in favor of an alternative hypothesis.

Alternative Hypothesis

Contrary to the null hypothesis, the alternative hypothesis, symbolized as H_a: μ>80, represents a statement that there is an effect or a difference—in this case, that the mean arsenic level is greater than 80 ppb. When conducting a hypothesis test, researchers look for evidence in their sample data that supports this hypothesis. It reflects the outcome that the study aims to prove. For the restaurant chain, proving the alternative hypothesis would mean finding significant evidence that a supplier's chicken meat has higher than acceptable levels of arsenic. The rejection of a null hypothesis leads to acceptance of the alternative hypothesis, prompting further action.

Statistical Significance

Statistical significance is all about determining the robustness of the results. In hypothesis testing, it relates to the probability of observing the given data—or something more extreme—if the null hypothesis were true. It is assessed using a p-value; a lower p-value suggests that the observed data is less likely to occur if the null hypothesis were true. When the p-value is less than a predetermined significance level—often set at 0.05 or 5%—the result is considered statistically significant. This means there's strong evidence against the null hypothesis, leading to its rejection. When results are statistically significant, as with Supplier A's p-value of 0.0003, the data suggests that the mean arsenic level is indeed greater than 80 ppb.

Mean Arsenic Level

The mean arsenic level in this context refers to the average amount of arsenic found in chicken meat, measured in parts per billion (ppb). This figure is critical for assessing food safety and consumer health. Regulatory or health organizations often set acceptable levels, and any mean level above this standard could lead to health risks. In the given exercise, the concern is whether the mean arsenic level exceeds 80 ppb. Accurate determination of this average is essential not only for ensuring public health but also for suppliers to maintain their business relationship with the restaurant chain.

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. In the provided scenario, two suppliers are being evaluated to determine if their chicken meat has an average arsenic level above 80 ppb. Various steps are involved, including stating the hypotheses, choosing an appropriate test, calculating the p-value, comparing it to the significance level, and making a conclusion. The evidence from both suppliers leads to different conclusions; Supplier A shows strong evidence against the null hypothesis, while Supplier B does not. Such a testing process is vital for making informed decisions based on data rather than assumptions.