Question

After a number of complaints about its directory assistance, a telephone company examined sam- ples of calls to determine the frequency of wrong numbers given to callers. Each sample consisted of 100 calls. Determine 95 percent limits. Is the process stable (i.e., in control)? Explain.

Answer 1

To assess the stability of the telephone company's directory assistance, one must calculate a 95 percent confidence interval for the proportion of wrong numbers using sample data. The critical value from the z-distribution and the sample proportion are needed to find the confidence limits. Stability is indicated by consistent confidence intervals across different samples.

Step-by-step explanation:

To determine the 95 percent confidence limits for the frequency of wrong numbers given to callers, one would use the methodology for constructing a confidence interval for a proportion. First, the percentage of wrong numbers in a sample must be calculated. This will serve as the sample proportion p'. Then, using the appropriate formula, calculate the confidence interval by taking p' plus or minus the margin of error, where the margin of error is the product of the critical value from the z-distribution (which corresponds to the 95% confidence level) and the standard error of the proportion.

The process is considered stable or 'in control' if the confidence intervals constructed from different samples do not vary significantly, meaning that the proportion of wrong numbers given out does not fluctuate beyond what is expected by random sampling variability. However, without specific data provided from the samples (e.g., the number of wrong numbers per 100 calls in each sample), one cannot definitively determine whether the process is stable.

Answer 2

Step-by-step explanation:

SAMPLE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Number of Errors 5 3 5 7 4 6 8 4 5 9 3 4 5 8 6 7

Average Defectives calculated by

This generates 87/ 16 (100) = 0.54. In order to obtain control limits, we apply the formula

Implying that

= 0.54 ± 0.044, this indicates that

The upper control limit = 0.54 + 0.044 = 0.10 (following conversion into fraction forms)

The lower control limit = 0.54 – 0.044 = 0.01 (following conversion into fraction forms)

From the calculation and table above, it is ideal to conclude that the process is out of control. This is because ¾ of the values appear to be above 0.04 despite all values falling within the limits.