If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Interpreting confidence levels and confidence intervals

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?
Choose all answers that apply:

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)?
Choose all answers that apply:

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?
Choose 1 answer:

Want to join the conversation?

  • blobby green style avatar for user Ann.T.Sebastian
    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?
    (26 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Sarah Ackerman
      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%).
      (78 votes)
  • winston default style avatar for user Abbas Al-bayati
    What is this exactly ?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • leafers tree style avatar for user Jared Bailey
    In the try it for yourself exercise, what are the blue and red curves??
    (7 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user B1-66ER
      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.
      (5 votes)
  • blobby green style avatar for user owen-k
    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'?
    (7 votes)
    Default Khan Academy avatar avatar for user
  • hopper cool style avatar for user Mark Ionkin
    Why don't we use a 100% confidence interval?
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user keetolia000
    us 100% interval?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • boggle purple style avatar for user Simran
      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
      (6 votes)
  • duskpin seed style avatar for user Balaji Harihar
    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
    (4 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Chuck B
      This caught me off guard at first as well. However, it's actually so simple no proof is needed.

      How far is a from b? It is |a-b|. How far is b from a? It is |b-a|, which is equal.

      When the sample mean falls within 2 standard deviations of the population mean, it is therefore equivalent to say that the population mean is within 2 standard deviations of the sample mean. That one occurs with 95% probability necessarily results in the other occurring with equal probability.

      It's important to note that in this context, "standard deviation" refers to a single value: the standard deviation of the population. We're not, for example, saying that "there is a 95% probability that the sample mean falls within 2 population standard deviations of the population mean" is equivalent to "There is a 95% probability that the population mean falls within 2 sample standard deviations of the sample mean."
      (1 vote)
  • blobby green style avatar for user grace9570
    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.
    (3 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user jayceelagula
    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?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Misha Fathi
    can you explain me this question please?


    A random sample of 32 persons attending a certain diet clinic was found to have lost, over a threeweek period, an average of 30 pounds with a sample standard deviation of 11 pounds. Construct a 99% confidence interval estimate of the true mean weight loss, over a three-week period, experienced by all persons attending the clinic. You may assume that the distribution of weight loss is normal.
    (1 vote)
    Default Khan Academy avatar avatar for user