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## Statistics and probability

### Course: Statistics and probability>Unit 11

Lesson 1: Introduction to confidence intervals

# 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 $90\mathrm{%}$ confidence interval for the true proportion of all voters who support the candidate.
Which of the following is a correct interpretation of the $90\mathrm{%}$ confidence level?

## 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 $95\mathrm{%}$ confidence interval for the mean speed. The resulting interval was $\left(110,120\right)$.
Which of the following is a correct interpretation of the interval $\left(110,120\right)$?

## 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 $99\mathrm{%}$ confidence level.
How will increasing the confidence level from $95\mathrm{%}$ to $99\mathrm{%}$ affect the confidence interval?

## Want to join the conversation?

• 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%).
• What is this exactly ?
• Math
• 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.
• us 100% interval?
• 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
• 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