Cancer In Young Children Is Usually Thought To Be Rare

cancer In Young Children Is Usually Thought To Be Rare It Is Belie

Cancer in young children is usually thought to be rare. It is believed that there is a 4% probability that a 0 to 4 year old will contract some form of the disease; however, you fear that your town may be a “cancer cluster” (a location with higher than normal incidences of cancer). In order to test this hypothesis, you gathered data on 100 children between the ages of 0 and 4 over the past 7 years. Four of these children have been diagnosed with cancer. Is your neighborhood a “cancer cluster”?

Paper For Above instruction

The concern about whether a specific geographic location exhibits a higher incidence of cancer than expected is a significant epidemiological question. Determining if a neighborhood qualifies as a “cancer cluster” involves statistical analysis to compare observed cases with the expected number based on prior probabilities. Here, the data comprises observations of 100 children aged 0 to 4 over seven years, with 4 diagnosed cases of cancer, against an expected probability of 4% (or 0.04) for each child to develop cancer. This scenario warrants a hypothesis test to evaluate whether the observed data deviates significantly from what would be expected under normal circumstances.

The hypotheses are formulated to test if the observed number of cancer cases exceeds what would be anticipated by chance. Specifically:

• Null hypothesis (H₀): The neighborhood is not a cancer cluster; the observed number of cases is consistent with the expected probability (p = 0.04).
• Alternative hypothesis (H_a): The neighborhood is a cancer cluster; the observed number of cases is greater than what would be expected (p > 0.04).

Based on the data, the number of observed cases is 4 out of 100 children. The expected number of cases under the null hypothesis is 100 * 0.04 = 4. This presents a situation suitable for a binomial test, but given the sample size, a normal approximation to the binomial distribution can be employed for simplicity.

Calculating the Z-Score

The z-score measures how many standard deviations the observed value (4 cases) is from the expected value (4 cases). The standard deviation (σ) of a binomial distribution is calculated as:

σ = √(n p (1 - p)) = √(100 0.04 0.96) = √(3.84) ≈ 1.96

The z-score is then determined by:

z = (Observed - Expected) / σ = (4 - 4) / 1.96 = 0 / 1.96 = 0

Calculating the p-Value

Since the z-score is 0, the corresponding p-value for a one-tailed test (testing if the observed is significantly greater than expected) is 0.5, indicating a 50% probability of observing 4 or fewer cases if the true rate is 4%. However, because the test is one-sided, we are primarily interested in the probability of observing more than 4 cases.

Given the z-value of 0, the p-value for the upper tail (greater than observed) is also 0.5.

Hypothesis Testing and Conclusion

Since the p-value (0.5) is much higher than the typical significance level of 0.05, we fail to reject the null hypothesis. This indicates that there is no statistically significant evidence to suggest that the neighborhood has a higher incidence of cancer than expected by chance.

The conclusion is that, based on this sample data, the observed number of cancer cases does not support the hypothesis that the neighborhood is a cancer cluster. It is consistent with the expected rate, and there is no statistical basis to claim an elevated risk.

Interpretation of Results

The analysis demonstrates that a single occurrence of 4 cases in 100 children, given a 4% expected rate, does not represent a significant deviation from normal expectations. It underscores the importance of larger sample sizes or additional data to detect any true increase in cancer incidence. Public health investigations often require more extensive data collection over time and across different areas to establish reliable patterns. Therefore, the current evidence does not warrant classification of this neighborhood as a cancer cluster but highlights the need for ongoing surveillance and research into environmental or genetic factors that may influence childhood cancer risks.

References

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