Final answer:
The upper control limit (UCLp) for the p-chart is 0.9821 (but adjusted to 1 since a proportion cannot exceed 1), and the lower control limit (LCLp) is 0.4379.
Step-by-step explanation:
To determine the upper and lower control limits (UCL and LCL) for the p-chart, we first need to calculate the average proportion of correct responses and its standard deviation. Since the question informs us that historically the average proportion of correct responses has been 71%, we denote this as p = 0.71. Next, we find the standard deviation for the proportion using the formula σp = √p(1 - p)/n, where n is the sample size, which in this case is 25 (the number of questions).
Let's calculate the standard deviation for the proportion:
σp = √(0.71)(1-0.71)/25 = √(0.71)(0.29)/25 = √0.2059/25 = √0.008236
σp = 0.0907
Now, using z = 3 for the 3-sigma control limits, the UCL and LCL can be calculated as follows:
UCLp = p + zσp = 0.71 + (3)(0.0907) = 0.71 + 0.2721 = 0.9821
LCLp = p - zσp = 0.71 - (3)(0.0907) = 0.71 - 0.2721 = 0.4379
However, since a proportion cannot exceed 1, we adjust the UCL to 1.
The Upper Control Limit (UCLp) is 0.9821 (adjusted to 1) and the Lower Control Limit (LCLp) is 0.4379.