Math Problem Statement

  1. A call center has 30 operators. Management wants to determine the proportion of time that the operators are idle. A work sampling study was carried out and 20 observations were made. The results are the following: Observation No. Number of operators idle Observation No. Number of operators idle 1 4 11 4 2 5 12 3 3 7 13 6 4 6 14 3 5 5 15 7 6 5 16 4 7 6 17 7 8 4 18 6 9 2 19 5 10 3 20 5 If the management wants an accuracy of 2% with a confidence level of 98%, what should be the number of sample observations taken?

Solution

To calculate the number of sample observations required for the desired accuracy and confidence level, we use the formula for the sample size in proportion estimation:

n=Z2p(1p)E2n = \frac{Z^2 \cdot p \cdot (1-p)}{E^2}

Where:

  • ZZ: The Z-score corresponding to the confidence level.
  • pp: The sample proportion (proportion of idle time in this case).
  • EE: The margin of error (desired accuracy).

Step 1: Calculate the sample proportion (pp):

p=Total idle observationsTotal operators across all observationsp = \frac{\text{Total idle observations}}{\text{Total operators across all observations}}

Total idle observations = Sum of idle operators from all observations: 4+5+7+6+5+5+6+4+2+3+4+3+6+3+7+4+7+6+5+5=1034 + 5 + 7 + 6 + 5 + 5 + 6 + 4 + 2 + 3 + 4 + 3 + 6 + 3 + 7 + 4 + 7 + 6 + 5 + 5 = 103

Total operators across all observations = 20×30=60020 \times 30 = 600.

p=1036000.1717p = \frac{103}{600} \approx 0.1717

Step 2: Determine the Z-score for a 98% confidence level:

The Z-score corresponding to a 98% confidence level is approximately Z=2.33Z = 2.33.

Step 3: Use the margin of error (EE):

The desired accuracy is 2%, so E=0.02E = 0.02.

Step 4: Calculate nn:

Substitute values into the formula: n=2.3320.1717(10.1717)0.022n = \frac{2.33^2 \cdot 0.1717 \cdot (1 - 0.1717)}{0.02^2}

First, calculate the numerator: 2.332=5.4289,0.1717(10.1717)=0.17170.8283=0.14222.33^2 = 5.4289,\quad 0.1717 \cdot (1 - 0.1717) = 0.1717 \cdot 0.8283 = 0.1422

Numerator=5.42890.14220.7715\text{Numerator} = 5.4289 \cdot 0.1422 \approx 0.7715

Now calculate the denominator: Denominator=0.022=0.0004\text{Denominator} = 0.02^2 = 0.0004

n=0.77150.00041928.75n = \frac{0.7715}{0.0004} \approx 1928.75

Final Answer:

The management should take at least 1929 observations to achieve an accuracy of 2% with a confidence level of 98%.


Let me know if you want additional clarifications or a breakdown of any step.
Here are five related questions:

  1. How do we interpret the Z-score in the context of confidence intervals?
  2. Why is it important to calculate the sample size before conducting a study?
  3. How would increasing the confidence level affect the required sample size?
  4. What role does the margin of error play in determining sample size?
  5. How can we estimate the proportion if no prior study data is available?

Tip: When estimating proportions without prior data, use ( p = 0.

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Math Problem Analysis

Mathematical Concepts

Proportion Estimation
Confidence Intervals
Sample Size Determination

Formulas

n = (Z^2 * p * (1 - p)) / E^2
p = Total idle observations / Total operators across all observations

Theorems

Central Limit Theorem (underpins confidence intervals)

Suitable Grade Level

Grades 11-12, College Level