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?

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%.

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?

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.

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?

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.

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

assuming you meant positive 8 it would equal out to 8 +/- 2.97

Main content

Statistics and probability

Interpreting a confidence interval for a mean

Google Classroom

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

\bar{x} = 148

microns and a standard deviation of

s_{x} = 3.3

microns.

95 %

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

\bar{x} = 31.8

and the standard deviation was

s_{x} = 4.3

years. A

95 %

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

\bar{x} = 394 ppm

and a standard deviation of

s_{x} = 26.3 ppm

. The resulting

95 %

confidence interval for the mean lead level is

394 \pm 9.8 .

What does this interval suggest?

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 %

confidence interval for the mean charging time. The resulting interval was

124 \pm 2.24

minutes.

What does this interval suggest about the charging times?
Hint: The specification was "no more than $2$ ‍ hours."

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007
Posted 5 years ago. Direct link to 007's post “a random sample of 36 dri...”
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?
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(1 vote)
Answer
- Jerry Nilsson
  Posted 5 years ago. Direct link to Jerry Nilsson's post “The conditions for infere...”
  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.
  Comment on Jerry Nilsson's post “The conditions for infere...”
  (6 votes)
javidjamae
Posted 3 years ago. Direct link to javidjamae's post “The wording on Example 1 ...”
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.
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(3 votes)
Answer
- Victor Gutierrez
  Posted 3 years ago. Direct link to Victor Gutierrez's post “I think the problem is co...”
  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.
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  (1 vote)
Nick Duran
Posted 2 years ago. Direct link to Nick Duran's post “So, like, I still have tr...”
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?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Mal
  Posted 2 years ago. Direct link to Mal's post “It's a convention. You wo...”
  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%.
  Button navigates to signup page
  (3 votes)
BootesVoidPointer
Posted 3 years ago. Direct link to BootesVoidPointer's post “In the examples where the...”
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?
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(1 vote)
Answer
- Victor Gutierrez
  Posted 3 years ago. Direct link to Victor Gutierrez's post “you could do that to prov...”
  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.
  Button navigates to signup page
  (2 votes)
Naresh K
Posted 3 years ago. Direct link to Naresh K's post “Are all plausible values ...”
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.
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(1 vote)
Answer
emily
Posted 3 years ago. Direct link to emily's post “I'm a bit confused. In pr...”
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?
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(1 vote)
Answer
younes.alexey
Posted 4 years ago. Direct link to younes.alexey's post “Two hundred students are ...”
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)
Button navigates to signup pageComment on younes.alexey's post “Two hundred students are ...”
(1 vote)
Answer
Nick Duran
Posted 2 years ago. Direct link to Nick Duran's post “A study was completed in ...”
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?
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(1 vote)
Answer
Lexline Johnson
Posted 2 years ago. Direct link to Lexline Johnson's post “Would we still be able to...”
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 😅
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(1 vote)
Answer
- William Wang
  Posted 2 years ago. Direct link to William Wang's post “It's too bad that I didn'...”
  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.
  Comment on William Wang's post “It's too bad that I didn'...”
  (1 vote)
Andrew Chantiloupe
Posted 5 years ago. Direct link to Andrew Chantiloupe's post “in order to monitor the w...”
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
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(0 votes)
Answer
- devinesungsoo
  Posted 5 years ago. Direct link to devinesungsoo's post “assuming you meant positi...”
  assuming you meant positive 8 it would equal out to 8 +/- 2.97
  Button navigates to signup page
  (2 votes)