Final answer:
The 95% confidence interval for the true proportion of the die landing on five dots, after observing it land on five 25 out of 100 times, is between 0.165 and 0.335. The closest range offered in the options is 0.15 to 0.25, which is Option C.
Step-by-step explanation:
A student has observed a die that landed on five dots 25 times after rolling it 100 times. To determine the confidence interval for the true proportion of this die landing on five, we can use the formula for a proportion confidence interval. The sample proportion (p-hat) is 25/100 = 0.25. At a 95% confidence level, the Z-score corresponding to this confidence level is approximately 1.96 for a two-sided interval. Using the standard error (SE) for a proportion which is sqrt(p-hat*(1-p-hat)/n), we find SE to be sqrt(0.25*0.75/100) = 0.0433. The margin of error (ME) is Z*SE, which gives us 1.96*0.0433 = 0.085. Adding and subtracting the margin of error from the sample proportion yields the confidence interval: 0.25 - 0.085 to 0.25 + 0.085, which is 0.165 to 0.335.
To provide a reasonable range that includes the observed proportion and fits within one of the choices given, the closest range that covers this interval is Option C, which is 0.15 to 0.25. Thus, for the true proportion of times the die will land on five dots at a 95% confidence level, a reasonable range would be Option C (0.15 to 0.25).