Question

The IRS is concerned with improving the accuracy of tax info given by its reps over the phone. Previous studies involved asking a set of 25 questions of a large number of IRS telephone reps to determine the proportion of correct responses. Historically, the average proportion of correct responses has been 71%. Recently, the set of 25 questions were again asked of 20 randomly selected reps. The proportions of correct answers were:

15, 15, 19, 17, 15, 17, 21, 13, 17, 13, 24, 16, 23, 16, 21, 20, 23, 19, 21, and 16.

A) What are the upper and lower control limits for the appropriate p-chart for the IRS? Use z = 3

The UCLp = ___

The LCLp = ___

Answer 1

The upper and lower control limits for the p-chart are determined by calculating the average proportion of correct responses, using it to compute the standard deviation for proportions, and applying the z-value of 3 to find the UCLp and LCLp.

Step-by-step explanation:

To calculate the upper and lower control limits (UCL and LCL) for the p-chart, you would first need to find the average proportion (π) of correct responses from the sample and then use the standard deviation for proportions to calculate the limits using the given z-value of 3 (which corresponds to 99.7% confidence level in control chart analysis).

The formulae for the UCLp and LCLp would be:

• UCLp = π + z * sqrt[(π(1-π))/n]
• LCLp = π - z * sqrt[(π(1-π))/n]

Where π is the average proportion, z is the z-value, and n is the sample size for each proportion measurement (in this case, the number of questions, 25).

First, calculate the average proportion (π) by summing the number of correct responses and dividing by the total number of responses. Then, substitute the values into the UCLp and LCLp formulas to find your control limits.

For this data set:

1. Calculate π by summing the given proportions and dividing by the number of observations.
2. Calculate the standard deviation using π and the number of questions per rep (n = 25).
3. Calculate UCLp and LCLp using the z-value of 3.

Note: Since the proportions are given as numbers of correct responses out of 25, they should be converted into proportions by dividing by 25 before using them in calculations.

Answer 2

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.