Final answer:
The authors can be 95% confident that the true proportion of teens who don't eat enough fruits and vegetables is between a calculated lower and upper bound using the margin of error. Factors beyond the margin of error that could affect survey outcomes include question phrasing, survey mode, nonresponse bias, and sampling frame. Decreasing the confidence level from 99% to 90% would result in a narrower confidence interval.
Step-by-step explanation:
The student's question concerns the analysis of survey data and the calculation of confidence intervals for proportions, which is a topic in statistics, a branch of mathematics. Given the information that 138 out of 230 teens did not eat enough fruits and vegetables and the margin of error is 6.3% for a 95% confidence level, the claim that the authors can include in their report is:
We are approximately 95% confident that the true proportion of teens who don't eat enough fruits and vegetables is between the lower bound and the upper bound. To calculate the bounds, we use the sample proportion (p), which is 138/230, and the margin of error (E), which is 6.3% of the proportion. The confidence interval is then calculated as (p - E, p + E).
Factors Affecting Survey Outcomes Beyond Margin of Error
The survey's outcome can be affected by several factors not covered by the margin of error such as:
- Question phrasing and order
- Survey mode effects (telephone vs. in-person)
- Nonresponse bias
- Population representation (sampling frame)
Without performing actual calculations, if the confidence level decreased from 99% to 90%, the width of the confidence interval would also decrease, making it narrower. This is because a lower confidence level means we require less certainty about our estimate, which allows for a smaller interval around the sample proportion to capture the true population proportion.
In the case of surveying the amount of fruit available in school lunches, a success would be a school that offers fruit in its lunches every day.
Confidence Intervals and Sample Size
If 40 heads of lettuce were sampled instead of 20, and either the error bound or the confidence level remained the same, the confidence interval would generally be expected to become narrower due to the increased sample size, which tends to give a more precise estimate of the population mean.