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A magazine provided results from a poll of 24 adults who were asked to identify their favorite pie. Among the respondents, 15 chose chocolate pie. If the confidence level is $98 \%$, calculate the confidence interval for the proportion of adults who identify chocolate pie as their favorite pie.
Enter the lower bound of the confidence interval as a percentage without the \% sign. (Round your percentage to the nearest tenth.)
Answer
Final Answer: The lower bound of the confidence interval for the proportion of adults who identify chocolate pie as their favorite pie, at a 98% confidence level, is \(\boxed{39.5\%}\).
Step 1 ：The question is asking for the lower bound of the confidence interval for the proportion of adults who identify chocolate pie as their favorite pie. To solve this, we can use the formula for the confidence interval for a proportion, which is given by: \(p \pm Z \sqrt{\frac{p(1-p)}{n}}\) where: p is the sample proportion (in this case, the proportion of adults who chose chocolate pie), n is the sample size (in this case, the number of adults polled), Z is the Z-score, which corresponds to the desired confidence level (in this case, 98%).
Step 2 ：First, we need to calculate the sample proportion (p), which is the number of adults who chose chocolate pie divided by the total number of adults polled. In this case, the number of adults who chose chocolate pie is 15 and the total number of adults polled is 24. So, \(p = \frac{15}{24} = 0.625\).
Step 3 ：Then, we need to find the Z-score that corresponds to a 98% confidence level. We can look this up in a standard normal distribution table. The Z-score for a 98% confidence level is approximately 2.326.
Step 4 ：Finally, we can substitute these values into the formula to calculate the confidence interval. The lower bound of the confidence interval is given by \(p - Z \times \sqrt{\frac{p(1-p)}{n}}\). Substituting the values we have, the lower bound is \(0.625 - 2.326 \times \sqrt{\frac{0.625(1-0.625)}{24}} = 0.395\).
Step 5 ：Converting this to a percentage and rounding to the nearest tenth, we get 39.5%.
Step 6 ：Final Answer: The lower bound of the confidence interval for the proportion of adults who identify chocolate pie as their favorite pie, at a 98% confidence level, is \(\boxed{39.5\%}\).
