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## Statistics and probability

### Course: Statistics and probability > Unit 11

Lesson 3: Estimating a population mean- Introduction to t statistics
- Simulation showing value of t statistic
- Conditions for valid t intervals
- Reference: Conditions for inference on a mean
- Conditions for a t interval for a mean
- Example finding critical t value
- Finding the critical value t* for a desired confidence level
- Example constructing a t interval for a mean
- Calculating a t interval for a mean
- Confidence interval for a mean with paired data
- Making a t interval for paired data
- Interpreting a confidence interval for a mean
- Sample size for a given margin of error 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 x, with, \bar, on top, equals, 148 microns and a standard deviation of s, start subscript, x, end subscript, equals, 3, point, 3 microns.

A 95, percent confidence interval for the mean thickness based on his data is left parenthesis, 147, point, 1, comma, 148, point, 9, right parenthesis.

**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, point, 1 and 148, point, 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 x, with, \bar, on top, equals, 31, point, 8 and the standard deviation was s, start subscript, x, end subscript, equals, 4, point, 3 years. A 95, percent confidence interval for the mean based on her data was left parenthesis, 30, point, 2, comma, 33, point, 4, right parenthesis.

**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 x, with, \bar, on top, equals, 394, start text, p, p, m, end text and a standard deviation of s, start subscript, x, end subscript, equals, 26, point, 3, start text, p, p, m, end text. The resulting 95, percent confidence interval for the mean lead level is 394, plus minus, 9, point, 8, point

## 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, percent confidence interval for the mean charging time. The resulting interval was 124, plus minus, 2, point, 24 minutes.

## Want to join the conversation?

- a random sample of 36 drivers used on average 749 gallons of gasoline per year. if the standard deviation of the population is 32 gallons, find the 95% confidence interval of the mean for all drivers. if a driver said that he used 803 gallons per year, would you believe that?(1 vote)
- The conditions for inference are met and so the confidence interval is

𝑥̅ ± 𝑧* ∙ 𝜎∕√𝑛 =

= 749 ± 1.96 ∙ 32∕√36 ≈

≈ (738, 760)

This means that we are 95% confident that the population mean is within this interval.

It doesn't tell us anything about the shape of the population distribution though.

Thereby we can't really tell how likely it is for someone to use 803 gallons of fuel per year and without other evidence there is no reason for us to doubt a driver who claims they use that much fuel.(6 votes)

- The wording on Example 1 is a bit confusing: "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."

In the first sentence I quoted, does "a certain part" describe the population mean across all instances of that "certain part"? Or, is it talking about the thickness of a specific instance of that part? In other words, the specificity of "certain part" is ambiguous.

Then in the second sentence it says "one of these parts". But, it is ambiguous what "these parts" is referring to. Is it referring to an instance of the type of part from the previous sentence, or is it generally talking about some part of the many parts the factory manufactures.

I would suggest using more specific words to make this more clear. Maybe: "The mean thickness on any point of any Part #1234 is 150 microns. Felix takes a single instance of Part #1234 to inspect and measures it in 50 randomly selected spots."

I'm not sure how to fix it, but it is very hard to read as it can be interpreted in several ways.(3 votes)- I think the problem is considering the population to be all the points in each one of the pieces manufactured. That is the only way I can make sense of it.(1 vote)

- So, like, I still have trouble with this... Where do we even get the 95, 90, 99% confidence? Do we just make it up? Just choose randomly, "I feel like a 97% today..." or what?(1 vote)
- It's a convention. You would be free to use what figure you like but it is commonplace/traditional to use a stat of 95% or 99%.(3 votes)

- In the examples where the population mean is in the confidence interval, do we not need to take multiple sets of samples to conclude that it is in the interval 95% of the time?(1 vote)
- you could do that to prove that the formulas work.. But it has been proven and you can safely assume the formulas work. Keep in mind you need to meet the requirements for your sample: randomness, normality and independence.(2 votes)

- Are all plausible values in the confidence interval equally likely. I am inclined to believe that the values in the middle of the confidence interval are more likely to be the true mean than those at the extremes. Since, those values outside of the confidence interval are considered not plausible, for instance if we question whether 44.9 or 55.1 are plausible true means in a sample giving confidence interval for mean as 45-55, the answer will be no. So, if 44.9 is not plausible and 55.1 is also not plausible, then logically 44 or 55 may not suddenly become plausible with the same likelihood for the entire interval. I am inclined to believe that they will be slightly more plausible than those values just outside the confidence interval, and the plausibility of values will increase as we approach the middle of the confidence interval. Can someone clarify this please? Thanks.(1 vote)
- I'm a bit confused. In previous lessons, it was specified that the confidence interval gave the likelihood (in%) that the real mean is contained in (x%) of the trials over numerous trials. Now it's saying the percentage gives you the likelihood that a specific interval contains the true percentage?(1 vote)
- Two hundred students are each asked to compute 95% CIs for a population mean based on data they will collect. What is the approximate probability that more than 190 of the CIs will cover the true mean? Please help me understand how to solve this.(The answer is 0.44)(1 vote)
- A study was completed in Florida. In southern Florida, the study involved 3,000 patients; 54% of them experienced flulike symptoms during the same month. The study had a margin of error of 1.8%. What does that mean for the study? I got the question right (The confidence interval is between 52.2% and 55.8%.) but I only guessed... I still don't understand it. My teacher also hasn't given a definite answer, just "The lessons will tell you..." but I keep checking the lessons and it still doesn't click in my head. Help please?(1 vote)
- Would we still be able to make the same conclusions if the confidence level were lower, for example 90%? I thought it would technically no longer meet the 0.05 default significance level, but I was told that the confidence level isn't too important and that we would still make the same conclusions?

Any last-minute insight before the ap exam would be greatly appreciated 😅(1 vote)- It's too bad that I didn't get to this before the exam came around! I hope that it went well for you.

The confidence level is important because it could work in conjunction with the idea of power and Type I/Type II errors.(1 vote)

- in order to monitor the weight of cows 6 randomly chosen the mean is -8.00 and standard deviation 2.828 construct 95% confidence interval(0 votes)
- assuming you meant positive 8 it would equal out to 8 +/- 2.97(2 votes)