Probability and Statistics > T-distribution > Confidence Interval For a Sample

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## Confidence Interval For a Sample: Overview

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

**Sample 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!

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I found this helpful but I am still not sure about the degrees of freedom. Even in the guided solution on the homework problems there are steps left out. It helps but I have to do alot of searching to understand where some of the numbers come from.

This sample really helped me a lot, because since I am not good at math it really helps that it is explained step by step not like math zone that its so confusing.

Mary,

Degrees of freedom is n-1,

Stephanie

proffesor I dont know what is happening but I understand how to do do this but when I do it this way on math zone it says its wrong and they show me how to do it a very different way, and I dont understand it.

Vanessa,

Can you send me an email with the Mathzone question you are working on? And your working out? That way, I’ll be able to compare them and see what’s going on.

Stephanie

I found your explanation very helpful. I was a little confused in step 3 where a=0.25, I thought it was 0.025. I think it might have been typed in error. Otherwise, everything else made sense.

I truly had no idea how to find the degrees of freedom, now from this site and again the step by step process I see I take the sample size and subtract 1. How easy is that!

Once again, these step by step instructions were very helpful.

Now this problem made since maybe its because it was a step by step process that I understood and it made a lot of sense regarding the sample size. So once I finished reading the instructions and started to work out the problem it all came to me. Which makes it a whole lot easier on my brain.

Like it. This is a very good example. Helps to have the table there for reference.

This was a life saver this week. The book and the CD do NOT give any help with this. Love that table, came in hand a lot this week.

This was really helpful I too was having problems understanding how to find and use the degrees of freedom, but thanks now I understand much better.

This was very helpful to me it explained by steps and I have it Thanks so Much

This site was very helpful to me and now I understand it very well!

I have a problem the the DF=85, how do I find the value on the table??

This was so helpful!

so that finally , what is the conclusion of calculation ? and what is the range of confidence ?

regards,

Omar

Civil Eng.

Omar,

See steps 6 and 7 for the CI,

Stephanie

Dear Stephanie,

Thank you very much for your reply.

regards,

Omar,

Like seriously this was so helpful… I’m a masscommunication student and I ad to do a course called quatitative techniques .. Had no idea about all this buh I took it on my self to learn it due to a reason dat im a programmer also buh it was challenging and it is still though.. I really got relieved after reading this HOW TO DO IT tips… Thanks a lot

Thanks, you helped simplify the problem I was having.

what is the t table value if df=59????

Rushad,

For which alpha value? It’ll be very close to the values at df=60.

If you have a TI-89 you can use that to get an exact answer:

How to find a t-distribution on the TI-89.

Stephanie

THANK YOU ANYWAY FOR THE INFORMATION BUT I AM REALLY GETTING CONFUSED ON HOW DOES MARGIN OF ERROR RELATE TO THE STANDARD DEVIATION

Hi, Nteka,

Could you post your question on our forum? One of our mods will be glad to answer your question.

Regards,

Stephanie

Thanks it was really helpful !

But why cant we straightaway Use Mean and Standard deviation instead of Standard error we have used in calculation above.

What is the cut off at which we can consider a sample a good reprensative of mean – where we can suitable use Mean +/- SD for calculations ?

Thnks in advance

Amanpreet,

Thanks for stopping by. Can you post your question in our forums? Click the Forums tab at the top ^^.

Thanks,

Stephanie