Monitoring phone calls to a toll-free number. A large...

Monitoring phone calls to a toll-free number. A large food-products company receives about 100,000 phone calls a year from consumers on its toll-free number. A computer monitors and records how many rings it takes for an operator to answer, how much time each caller spends "on hold," and other data. However, the reliability of the monitoring system has been called into question by the operators and their labor union. As a check on the computer system, approximately how many calls should be manually monitored during the next year to estimate the true mean time that callers spend on hold to within 3 seconds with $95 \%$ confidence? Answer this question for the following values of the standard deviation of waiting times (in seconds) 10,20 , and 30 .

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. In the context of monitoring phone calls, a confidence interval for the mean waiting time allows us to estimate the true average time with a specified level of certainty (e.g., 95%).

Margin of Error

The margin of error quantifies the maximum expected difference between the true population parameter and a sample estimate, given a particular confidence level. It is a critical component in designing studies because it determines the precision of the estimate, such as requiring the mean waiting time to be within 3 seconds of the true value.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. In this scenario, different assumed values for the standard deviation of waiting times (10, 20, and 30 seconds) are used to understand how variability impacts the sample size needed to reach the desired precision.

Sample Size Determination

Sample size determination involves calculating the number of observations required to estimate a population parameter with a desired precision and confidence level. It depends on the standard deviation of the population, the desired margin of error, and the critical value corresponding to the selected confidence level (e.g., 1.96 for 95% confidence).

Transcript

00:01 So at a 5 % significance level, we want to find whether the meantime for software repair is equal to that for hardware repair for this company, or that the software takes longer on the average than the hardware to repair.

00:19 And we are going to be using a z value.

00:21 We'll see why in a moment.

00:23 So let's find what that critical value is to have 5 % in that upper tail, and it will be a z.

00:30 Value since we're doing a one -tail test and that value will correspond with positive 1 .645.

00:37 So if we are higher than that value, we will be rejecting the null.

00:43 So let's get our data.

00:45 And the data we have is for the software and for the hardware.

00:49 And we have the mean for the length of time it takes to solve that issue or fix that problem.

00:59 Is 18 minutes with a standard deviation from the sample of 4 .2 and that sample size was 35.

01:06 And for the hardware, the hardware took on the average 15 .5 minutes, so it is lower, but is it significantly lower? and the sample standard deviation for that group was 3 .9 minutes and the sample size was 45.

01:21 And since again, both of these are greater than are equal to 30, our text tells us that a normal distribution is fine to use as the tested day because we have sample standard deviations and we normally think got to use a t but those sample sizes are sufficiently large for us to calculate a z value so let's see what our z value is we're going to take that 18 minus the 15 .5 and then we will find our standard air so we'll take our sample standard deviation squared divided by the sample size and our sample standard deviation square divided by the sample size.

02:01 And that gives us a test statistic of, whoops, let me quick click those buttons.

02:09 We get 2 .72...

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