Question 3 : How will increasing the confidence level from 95 percent to 99 percent affect the confidence interval? In the above mentioned question, wouldn't the interval be narrower. If we were to decrease the confidence level to say 85%, the margin of error will be more and the interval will be wider?

The wider your interval is, the more confident you can be that your interval contains the true mean. Think about an interval that covers the entire spread of the data... you can be 100% confident that it contains the true mean. If you have a very narrow range (e.g. 115.1 mph to 115.2 mph), then in this example you cannot be very confident that it will contain the true mean, so you'd have a very low confidence interval (near 0%).

In the try it for yourself exercise, what are the blue and red curves??

The red line is the sample distribution, the blue line is the population distribution. The reason why the blue line changing it's shape while adjusting sample size is the scale of the whole chart is changing, which means the blue line actually isn't changing at all, just the zooming out to in order to show the full area of red line, I think.

Drawing more samples causes the interval to narrow, lowering the confidence level also causes the confidence interval to narrow. But I think a 99% confidence level means you are more certain that your population parameter would fall into that interval, right? So why is a narrow interval in terms of a higher sample size 'good' but a narrow interval in terms of a smaller confidence level 'bad'?

Great question! When you increase the sample size "n", the Margin of error decreases. This is because the formula for Margin of Error (in proportions) is the critical value times the standard error. The standard error is sqrt (phat)(1-phat)/n, where n is the sample size. So, as you increase n which is in the denominator, the standard error decreases, which means that the margin of error decreases. A link which I found helpful is https://www.dummies.com/education/math/statistics/how-sample-size-affects-the-margin-of-error/

Why don't we use a 100% confidence interval?

The normal distribution is defined from negative infinity to positive infinity and the corresponding 100% confidence interval would be from negative infinity to positive infinity as well. It doesn't provide useful information, and thus it is not used.

This question was asked sometime ago (by Mark Ionkin), and Soo Kyung Ahn responded: "The normal distribution is defined from negative infinity to positive infinity and the corresponding 100% confidence interval would be from negative infinity to positive infinity as well. It doesn't provide useful information, and thus it is not used." Elias Aquino also said: "Imagine if weather reporters said "there is a 100% chance the weather today will be between -100 to 300 degrees" (useless info, not specific enough) versus, "we are 95% certain the weather will be between 20 and 38 degrees" VERSUS "we are 80% certain the weather will be between 29 and 32 degrees" (more specific, but less certain)." Omar Khalaf said: "having a 100% confidence interval will give us an interval from (-infinity, +infinity) since, technically, we can be a 100% sure that the value we want will be a number between these values. You can never be a 100% sure of something." So basically the confidence interval would have to include all numbers, which makes it kinda useless, right? Therefore, people typically don't use 100% confidence intervals. Hope that helps this is sorta new to me too

Main content

AP®︎/College Statistics

Course: AP®︎/College Statistics > Unit 10

Lesson 1: Introduction to confidence intervals

Interpreting confidence levels and confidence intervals

Google Classroom

When we create a confidence interval, it's important to be able to interpret the meaning of the confidence level we used and the interval that was obtained.

The confidence level refers to the long-term success rate of the method, that is, how often this type of interval will capture the parameter of interest.

A specific confidence interval gives a range of plausible values for the parameter of interest.

Let's look at a few examples that demonstrate how to interpret confidence levels and confidence intervals.

Example 1: Interpreting a confidence level

A political pollster plans to ask a random sample of

500

voters whether or not they support the incumbent candidate. The pollster will take the results of the sample and construct a

90 %

confidence interval for the true proportion of all voters who support the candidate.

Which of the following is a correct interpretation of the $90 %$ ‍ confidence level?

(Choice A)
If the pollster repeats this process and constructs $20$ ‍ intervals from separate independent samples, we can expect about $18$ ‍ of those intervals to contain the true proportion of voters who support the candidate.
(Choice B)
About $90 %$ ‍ of people who support the candidate will respond to the poll.
(Choice C)
If the pollster repeats this process many times, then about $90 %$ ‍ of the intervals produced will capture the true proportion of voters who support the candidate.

Example 2: Interpreting a confidence interval

A baseball coach was curious about the true mean speed of fastball pitches in his league. The coach recorded the speed in kilometers per hour of each fastball in a random sample of

100

pitches and constructed a

95 %

confidence interval for the mean speed. The resulting interval was

(110, 120)

Which of the following is a correct interpretation of the interval $(110, 120)$ ‍?

(Choice A)
If the coach took another sample of $100$ ‍ pitches, there's a $95 %$ ‍ chance the sample mean would be between $110$ ‍ and $120 km/hr$ ‍.
(Choice B)
About $95 %$ ‍ of pitches in the sample were between $110$ ‍ and $120 km/hr$ ‍.
(Choice C)
We're $95 %$ ‍ confident that the interval $(110, 120)$ ‍ captured the true mean pitch speed.

[Can we say there is a 95% chance that the true mean is between 110 and 120 kilometers per hour?]

Example 3: Effect of changing confidence level

Suppose that the coach from the previous example decides they want to be more confident. The coach uses the same sample data as before, but recalculates the confidence interval using a

99 %

confidence level.

How will increasing the confidence level from $95 %$ ‍ to $99 %$ ‍ affect the confidence interval?

(Choice A)
It's impossible to say without seeing the sample data.
(Choice B)
Increasing the confidence will increase the margin of error resulting in a wider interval.
(Choice C)
Increasing the confidence will decrease the margin of error resulting in a narrower interval.

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Ann.T.Sebastian
Posted 6 years ago. Direct link to Ann.T.Sebastian's post “Question 3 : How will inc...”
Question 3 : How will increasing the confidence level from 95 percent to 99 percent affect the confidence interval?

In the above mentioned question, wouldn't the interval be narrower.
If we were to decrease the confidence level to say 85%, the margin of error will be more and the interval will be wider?
Button navigates to signup pageComment on Ann.T.Sebastian's post “Question 3 : How will inc...”
(21 votes)
Answer
- Sarah Ackerman
  Posted 6 years ago. Direct link to Sarah Ackerman's post “The wider your interval i...”
  The wider your interval is, the more confident you can be that your interval contains the true mean. Think about an interval that covers the entire spread of the data... you can be 100% confident that it contains the true mean.
  If you have a very narrow range (e.g. 115.1 mph to 115.2 mph), then in this example you cannot be very confident that it will contain the true mean, so you'd have a very low confidence interval (near 0%).
  Comment on Sarah Ackerman's post “The wider your interval i...”
  (72 votes)
Abbas Al-bayati
Posted 4 years ago. Direct link to Abbas Al-bayati's post “What is this exactly ?”
What is this exactly ?
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(3 votes)
Answer
- david_v2347
  Posted 7 months ago. Direct link to david_v2347's post “Math”
  Math
  Comment on david_v2347's post “Math”
  (17 votes)
Jared Bailey
Posted 4 years ago. Direct link to Jared Bailey's post “In the try it for yoursel...”
In the try it for yourself exercise, what are the blue and red curves??
Button navigates to signup pageButton navigates to signup page
(7 votes)
Answer
- B1-66ER
  Posted 4 years ago. Direct link to B1-66ER's post “The red line is the sampl...”
  The red line is the sample distribution, the blue line is the population distribution. The reason why the blue line changing it's shape while adjusting sample size is the scale of the whole chart is changing, which means the blue line actually isn't changing at all, just the zooming out to in order to show the full area of red line, I think.
  Button navigates to signup page
  (4 votes)
owen-k
Posted 4 years ago. Direct link to owen-k's post “Drawing more samples caus...”
Drawing more samples causes the interval to narrow, lowering the confidence level also causes the confidence interval to narrow. But I think a 99% confidence level means you are more certain that your population parameter would fall into that interval, right? So why is a narrow interval in terms of a higher sample size 'good' but a narrow interval in terms of a smaller confidence level 'bad'?
Button navigates to signup pageButton navigates to signup page
(7 votes)
Answer
- vedant.joshi.208
  Posted 4 years ago. Direct link to vedant.joshi.208's post “Great question! When you ...”
  Great question! When you increase the sample size "n", the Margin of error decreases. This is because the formula for Margin of Error (in proportions) is the critical value times the standard error. The standard error is sqrt (phat)(1-phat)/n, where n is the sample size. So, as you increase n which is in the denominator, the standard error decreases, which means that the margin of error decreases. A link which I found helpful is https://www.dummies.com/education/math/statistics/how-sample-size-affects-the-margin-of-error/
  Comment on vedant.joshi.208's post “Great question! When you ...”
  (4 votes)
Mark Ionkin
Posted 4 years ago. Direct link to Mark Ionkin's post “Why don't we use a 100% c...”
Why don't we use a 100% confidence interval?
Button navigates to signup pageComment on Mark Ionkin's post “Why don't we use a 100% c...”
(3 votes)
Answer
- Soo Kyung Ahn
  Posted 4 years ago. Direct link to Soo Kyung Ahn's post “The normal distribution i...”
  The normal distribution is defined from negative infinity to positive infinity and the corresponding 100% confidence interval would be from negative infinity to positive infinity as well. It doesn't provide useful information, and thus it is not used.
  Button navigates to signup page
  (6 votes)
keetolia000
Posted 10 months ago. Direct link to keetolia000's post “us 100% interval?”
us 100% interval?
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(2 votes)
Answer
- Simran
  Posted 4 months ago. Direct link to Simran's post “This question was asked s...”
  This question was asked sometime ago (by Mark Ionkin), and Soo Kyung Ahn responded:
  
  "The normal distribution is defined from negative infinity to positive infinity and the corresponding 100% confidence interval would be from negative infinity to positive infinity as well. It doesn't provide useful information, and thus it is not used."
  
  Elias Aquino also said:
  "Imagine if weather reporters said "there is a 100% chance the weather today will be between -100 to 300 degrees" (useless info, not specific enough)
  
  versus, "we are 95% certain the weather will be between 20 and 38 degrees"
  
  VERSUS "we are 80% certain the weather will be between 29 and 32 degrees" (more specific, but less certain)."
  
  Omar Khalaf said:
  "having a 100% confidence interval will give us an interval from (-infinity, +infinity) since, technically, we can be a 100% sure that the value we want will be a number between these values. You can never be a 100% sure of something."
  
  So basically the confidence interval would have to include all numbers, which makes it kinda useless, right?
  
  Therefore, people typically don't use 100% confidence intervals.
  
  Hope that helps this is sorta new to me too
  Button navigates to signup page
  (5 votes)
Balaji Harihar
Posted 4 months ago. Direct link to Balaji Harihar's post “Is there a mathematical p...”
Is there a mathematical proof of the equivalence between the 2 statements:
Statement 1 - There is a 95% probability that the sample mean falls within 2 standard deviations of the population mean. Statement 2 - There is a 95% probability that the population mean falls within 2 standard deviations of the sample mean
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(3 votes)
Answer
grace9570
Posted 3 years ago. Direct link to grace9570's post “In Example 2, shouldn't B...”
In Example 2, shouldn't B also be correct? To construct the 95% confidence interval, we add/subtract 2 standard deviations from the mean. Given the distribution of the sample is approximately normal, this interval would also contain about 95% of the sample pitches.
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(3 votes)
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- zjleon2010
  Posted 3 years ago. Direct link to zjleon2010's post “recall how we calculate t...”
  recall how we calculate the standard deviation:
  sqrt(p*(1-p)/n)
  this is an estimation of the SD of the *population*
  so the confidence level it construct doesn't work for the sample data
  Button navigates to signup page
  (1 vote)
jayceelagula
Posted a year ago. Direct link to jayceelagula's post “So the best way to estima...”
So the best way to estimate the population mean is to set a higher confidence level AND increase the sample size if you want a good balance of how confident you are that the interval captures the true mean and how accurate that interval might be (i.e. low standard error) - is that correct?
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(1 vote)
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sassyshourya13
Posted 3 years ago. Direct link to sassyshourya13's post “What is the implication o...”
What is the implication of not being able to have 100% confidence mean to performing an analysis?
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(1 vote)
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