Probability and Statistics > Confidence Intervals > How to Find a Confidence Interval

**Contents**:

- How to Find a Confidence Interval for a Sample
- How to Find a Confidence Interval for a Proportion: Overview

## Confidence Interval For a Sample: Overview

Watch the video or read on below:

When you don’t know anything about a population’s behavior (i.e. you’re just looking at data for a sample), you need to use the **t-distribution** to find the **confidence interval**. That’s the vast majority of cases: you usually don’t know population parameters, otherwise you wouldn’t be looking at statistics!

The confidence interval tells you how confident you are in your results. With any survey or experiment, you’re never 100% sure that your results could be repeated. If you’re 95% sure, or 98% sure, that’s usually considered “good enough” in statistics. That percentage of sureness is the confidence interval.

## Confidence Interval For a Sample: Steps

**Question**:

A group of 10 foot surgery patients had a mean weight of 240 pounds. The sample standard deviation was 25 pounds. Find a confidence interval for a sample for the true mean weight of all foot surgery patients. Find a 95% CI.

**Step 1: ** *Subtract 1 from your sample size*. 10 – 1 = 9. This gives you degrees of freedom, which you’ll need in step 3.

**Step 2: ** *Subtract the confidence level from 1, then divide by two. *

(1 – .95) / 2 = .025

**Step 3: ** *Look up your answers to step 1 and 2 in the t-distribution table.* For 9 degrees of freedom (**df**) and **α = **0.025, my result is 2.262.

df |
α = 0.1 |
0.05 |
0.025 |
0.01 |
0.005 |
0.001 |
0.0005 |

∞ |
t_{α}=1.282 |
1.645 | 1.960 | 2.326 | 2.576 | 3.091 | 3.291 |

1 |
3.078 | 6.314 | 12.706 | 31.821 | 63.656 | 318.289 | 636.578 |

2 |
1.886 | 2.920 | 4.303 | 6.965 | 9.925 | 22.328 | 31.600 |

3 |
1.638 | 2.353 | 3.182 | 4.541 | 5.841 | 10.214 | 12.924 |

4 |
1.533 | 2.132 | 2.776 | 3.747 | 4.604 | 7.173 | 8.610 |

5 |
1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 5.894 | 6.869 |

6 |
1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 5.208 | 5.959 |

7 |
1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.785 | 5.408 |

8 |
1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 4.501 | 5.041 |

9 |
1.383 | 1.833 | 2.262 |

**Step 4: ***Divide your sample standard deviation by the square root of your sample size. *

25 / √(10) = 7.90569415

**Step 5: ** *Multiply step 3 by step 4.*

2.262 × 7.90569415 = 17.8826802

**Step 6: ***For the lower end of the range, subtract step 5 from the sample mean.*

240 – 17.8826802 = 222.117

**Step 7: ** *For the upper end of the range, add step 5 to the sample mean. *

240 + 17.8826802 = 257.883

That’s how to find the confidence interval for a sample!

**Like the explanation on how to find a confidence interval**? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.

**Questions**? Our Stats Guy can tackle your tricky stats questions on our 100% free forum!

## How to Find a Confidence Interval for a Proportion: Overview

When we talk about a confidence interval (CI), we’re dealing with data. For example, let’s say the manager for that job you applied for told you he would get back with you in a “couple of days.” A couple of days could mean two. Or three. Or there might be a paperwork backlog and it could be a week. It definitely doesn’t mean in an hour. So your CI would probably be between** 2 and 4 days.**

Perhaps the trickiest part of CIs is recognizing the various parts needed for the formula, like z a/2. This section breaks everything down into simple steps and shows you how to find a confidence interval for population proportions.

## How to Find a Confidence Interval for a Proportion: Steps

Question: 510 people applied to the Bachelor’s in Elementary Education program at Florida State College. Of those applicants, 57 were men. Find the 90% CI of the true proportion of men who applied to the program.

**Step 1:** *Read the question carefully and figure out the following variables:*

**α**: subtract the given CI from 1.

1-.9=.10**z**_{ α/2}: divide α by 2, then look up that area in the z-table. Not sure how to read a z table? See this article,**area to the right of a z-score**for an explanation on how to read the z-table.

.1/2=.0500. The closest z-value to an area of .0500 is 0.13- : Divide the proportion given (i.e. the smaller number)by the sample size.

57/510=0.112 - : Subtract from 1.

1-0.112 = 0.888

**Step 2:** *Multiply **by **.*

0.112 x 0.888 = 0.099456

**Step 3:** *Divide step 2 by the sample size*.

0.099456 / 510 = 0.000195011765

**Step 4:***Take the square root of step 3*:

sqrt(0.000195011765) = 0.0139646613

**Step 5:** *Multiply step 4 by z *_{a/2}*:*

0.0139646613 x 0.13 = 0.0182

**Step 6:**: *For the lower percentage, subtract step 5 from *.

0.112-0.0182 = 0.0938 = 9.38%

**Step 7:***For the upper percentage, add step 5 to *.

0.112 + 0.0182 = 0.1302 = 13.02%

That’s how to find a confidence interval!

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This step by step was a tremendous help to me!

Wow. I didn’t realize there were so many steps involved. This example helped alot because it broke everything up, step by step. Was helpful to have the link to the Z-table.

Yeah these problems have a lot of steps to them but once you get them down they are really easy because you have the Z table. And in order for you to do your problems you have to know how to use the tables. Which in a way I thought they were fun.

I can’t believe how much this helped me. I need the step by step explanation and this was great.

z α/2: divide the given confidence interval by 2, then look up that area in the z-table.

.9/2=.4500. The closest z-value to an area of .4500 is 1.65

I don’t understand this….can someone please explain?

What am I not getting?

Hi Jennifer,

Why don’t you send me your working out, and I’ll tell you where you are going wrong (or at least, point out which specific part you don’t understand…is it how to look up a number in the z-table? Or something else?)

Stephanie

The closest z-value to an area of .4500 is 1.65 …

This is where I am really stuck. How did you get that? I can look up .45 on the table and it does not come out to be 1.65

Jennifer,

What do you get when you look up .4500 in the center of the table? Let me know and I should be able to see where you are going wrong.

Sometimes it helps to work backwards…look up 1.65 in the table (i.e. 1.6 in the left hand column and .05 in the top row). Do you see the .4505 where it intersects?

Stephanie

Once again, I found this explanation very helpful. It was easier to follow your step by step instructions than the guided solution in mathzone.

The step by step instructions are very helpful. Thank you!

I have had the same question how do you get 1.65. But from this step by step and the teacher explaining to another student the same question, I see I am not the only one with these questions. This site is a big help!

I was wondering how to get confidence interval with standard deviation 0.3, confidence level of 90%, and satisfaction score of 4.5

Hi, Myrlande,

Can you post your question on the forum? Unfortunately, I don’t have the time to answer math questions here.

Thanks,

Stephanie

I have two z-tables in my text , for 1.65 it’s .9505 and for -1.65 it’s .0495, how does your table have .4505?

Bill,

I suspect you’re looking at a tenths and hundredths z-table, instead of the regular one. It’s common in biostatistics and other subjects where tiny fractions are needed.

You can confirm .4505 is correct (for the regular table) by going here, where they give some z-table examples: http://www.mhhe.com/socscience/crimjustice/stat-methods/book1/chap8.mhtml (question 2c).

Regards,

Stephanie

How do you get this?

The closest z-value to an area of .0500 is 0.13

Look up .0500 in the center of the z table. The closest you can get is at z=.13 (an area of .0517)