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### Course: Statistics and probability > Unit 11

Lesson 3: Estimating a population mean- Reference: Conditions for inference on a mean
- Conditions for a t interval for a mean
- Finding the critical value t* for a desired confidence level
- Calculating a t interval for a mean
- Making a t interval for paired data
- Interpreting a confidence interval for a mean
- Sample size and margin of error in a confidence interval for a mean

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# 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.

## Example 1

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 $150$ microns. Felix measured the thickness of $50$ randomly selected points on one of these parts to see if it was painted properly. His sample had a mean thickness of $\overline{x}=148$ microns and a standard deviation of ${s}_{x}=3.3$ microns.

A $95\mathrm{\%}$ confidence interval for the mean thickness based on his data is $(147.1,148.9)$ .

**Based on his interval, is it plausible that this part's average thickness agrees with the target value?**

No, it isn't. The interval says that the plausible values for the true mean thickness on this part are between $147.1$ and $148.9$ microns. Since this interval doesn't contain $150$ 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 $150$ microns, so this part's mean thickness is likely below the target.

## Example 2

Martina read that the average graduate student is $33$ years old. She wanted to estimate the mean age of graduate students at her large university, so she took a random sample of $30$ graduate students. She found that their mean age was $\overline{x}=31.8$ and the standard deviation was ${s}_{x}=4.3$ years. A $95\mathrm{\%}$ confidence interval for the mean based on her data was $(30.2,33.4)$ .

**Based on this interval, is it plausible that the mean age of all graduate students at her university is also**$33$ years?

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

## Example 3: Try it out!

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

These data show a sample mean of $\overline{x}=394{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{ppm}$ and a standard deviation of ${s}_{x}=26.3{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{ppm}$ . The resulting $95\mathrm{\%}$ confidence interval for the mean lead level is $394\pm 9.8.$

## Example 4: Try it out!

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

Sandra took a random sample of $40$ 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 $95\mathrm{\%}$ confidence interval for the mean charging time. The resulting interval was $124\pm 2.24$ minutes.

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