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In business, how can we measure the reliability of the things we create? The only foolproof way to accomplish this would be to test and analyze every single object produced, which is often impractical, if not impossible. Instead, we typically measure a small sample of objects and expect that this sample will be a representative of the entire population.
Any sample that we take for this purpose is only an approximation, and will always contain some amount of error or uncertainty. However, we can use the concept of confidence intervals to help determine how reliable we expect our sampling can be, compared to the entire population.
What Does the Confidence Interval Signify?
The confidence interval is a measure of the reliability of the sample mean compared to the actual mean. We can choose any confidence interval to express this information. If we choose 95% confidence, then the calculation says we can be 95% certain that the mean of the entire population will lie between the lower and upper range of the sample mean and standard deviation.
Another way to think about this is that if we ran through the sampling measurements 100 times, 95 of those results, expressed as the sample mean and range, are likely to contain the true mean. As you can see, this is not a guarantee of some specific range of results. Instead, confidence intervals express, in statistical terms, the confidence that our sample represents the entire population.
A Manufacturing Example
Motorola wishes to estimate the mean talk time for one of their new phones before the battery must be recharged. In a random sample of 35 phones, the sample mean talk time is 325 minutes.
A) Why can we say that the sampling distribution of x-bar is appropriately normal?
B) Construct a 98% Confidence Interval for the mean talk time for all new Motorola phones, assuming that the population standard deviation is 31 minutes.
C) Construct a 90% Confidence Interval for the mean talk time for all new Motorola phones, assuming that the population standard deviation is 31 minutes.
D) How many phones would Motorola need to test to estimate the mean talk time for all new Motorola phones within 5 minutes with a 99% confidence ?