Final answer:
To maintain a 96 percent confidence level with the same error bound, the firm will likely have to increase the sample size. Increasing the sample size helps in maintaining the error bound at a higher confidence level, whereas decreasing the sample size would diminish the confidence or increase the margin of error.
Step-by-step explanation:
When a firm decides it needs to be at least 96 percent confident of the population mean length of time to within one hour, the number of people the firm surveys might need to change. This is because as the level of confidence increases, the required sample size to estimate the population mean with a fixed margin of error also increases. This means that the firm will likely have to survey more people to achieve this higher level of confidence while keeping the error bound the same.
If the firm wished to increase its level of confidence and keep the error bound the same by taking another survey, it should increase the sample size. A larger sample size would decrease the standard error and thus maintain the same error bound.
Conversely, if the firm did another survey, kept the error bound the same, and only surveyed 49 people, the level of confidence would decrease. This is because a smaller sample size increases the standard error, which would then require either a larger margin of error or a lower confidence level to encompass the true population mean.
Success in statistical terms refers to the outcome of interest in a study. In the context of the state health board being concerned about the availability of fruit in school lunches, a success would be the case where schools do offer fruit every day, leading to a positive outcome for the board's concerns.