Final answer:
To calculate the prediction intervals for the proportion of red tickets, use the standard deviation of the sample proportion and apply Z-scores for the desired confidence levels, according to the Empirical Rule.
Step-by-step explanation:
You are asked to find the approximate prediction intervals for T, the proportion of red tickets sampled, under the assumption that the actual proportion of red tickets in the box (ϴ) is 2/5. Given that you sampled 200 tickets and found 57.5% red tickets, we can use the normal distribution to calculate the prediction intervals.
To find these intervals, we consider the standard deviation of the sample proportion which is given by the formula standard deviation of the sample proportion = √npq. Here n is the sample size, p is the proportion of red tickets according to the null hypothesis, and q is 1-p, the proportion of blue tickets.
Since the sample size is large (more than 100 words), the sampling distribution of T can be approximated by a normal distribution. We apply the Z-scores corresponding to the desired confidence levels: approximately 1 for 68%, 2 for 95%, and 3 for 99.7% (known as the Empirical Rule or the 68-95-99.7 rule). The prediction interval at each confidence level can be calculated by: predicted proportion = sampled proportion ± (Z-score * standard deviation of the sample proportion).