Interpreting a confidence interval for a mean

After we build a confidence interval for a mean, it's important to be able to interpret what the interval tells us about the population and what it doesn't tell us.

A confidence interval for a mean gives us a range of plausible values for the population mean. If a confidence interval does not include a particular value, we can say that it is not likely that the particular value is the true population mean. However, even if a particular value is within the interval, we shouldn't conclude that the population mean equals that specific value.

Let's look at few examples that demonstrate how to interpret a confidence interval for a mean.

Felix is a quality control expert at a factory that paints car parts. Their painting process consists of a primer coat, color coat, and clear coat. For a certain part, these layers have a combined target thickness of $150$150 microns. Felix measured the thickness of $50$50 randomly selected points on one of these parts to see if it was painted properly. His sample had a mean thickness of $\bar x=148$x, with, \bar, on top, equals, 148 microns and a standard deviation of $s_x=3.3$s, start subscript, x, end subscript, equals, 3, point, 3 microns.

A $95\%$95, percent confidence interval for the mean thickness based on his data is $(147.1,148.9)$left parenthesis, 147, point, 1, comma, 148, point, 9, right parenthesis.

No, it isn't. The interval says that the plausible values for the true mean thickness on this part are between $147.1$147, point, 1 and $148. 9$148, point, 9 microns. Since this interval doesn't contain $150$150 microns, it doesn't seem plausible that this part's average thickness agrees with the target value. In other words, the entire interval is below the target value of $150$150 microns, so this part's mean thickness is likely below the target.

Martina read that the average graduate student is $33$33 years old. She wanted to estimate the mean age of graduate students at her large university, so she took a random sample of $30$30 graduate students. She found that their mean age was $\bar x=31.8$x, with, \bar, on top, equals, 31, point, 8 and the standard deviation was $s_x=4.3$s, start subscript, x, end subscript, equals, 4, point, 3 years. A $95\%$95, percent confidence interval for the mean based on her data was $(30.2,33.4)$left parenthesis, 30, point, 2, comma, 33, point, 4, right parenthesis.

Yes. Since $33$33 is within the interval, it is a plausible value for the mean age of the entire population of graduate students at her university.

The Environmental Protection Agency (EPA) has standards and regulations that say that the lead level in soil cannot exceed the limit of $400$400 parts per million (ppm) in public play areas designed for children. Luke is an inspector, and he takes $30$30 randomly selected soil samples from a site where they are considering building a playground.

These data show a sample mean of $\bar x=394\,\text{ppm}$x, with, \bar, on top, equals, 394, start text, p, p, m, end text and a standard deviation of $s_x=26.3\,\text{ppm}$s, start subscript, x, end subscript, equals, 26, point, 3, start text, p, p, m, end text. The resulting $95\%$95, percent confidence interval for the mean lead level is $394 \pm 9.8.$394, plus minus, 9, point, 8, point

Sandra is an engineer working on wireless charging for a mobile phone manufacturer. Their design specifications say that it should take no more than $2$2 hours to completely charge a fully depleted battery.

Sandra took a random sample of $40$40 of these phones and chargers. She fully depleted their batteries and timed how long it took each of them to completely charge. Those measurements were used to construct a $95\%$95, percent confidence interval for the mean charging time. The resulting interval was $124 \pm 2.24$124, plus minus, 2, point, 24 minutes.