Statistics How To

Confidence Interval: How to Find it: The Easy Way!

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If you’re just beginning statistics, you’ll probably be finding confidence intervals using the normal distribution (see #3 below). But in reality, most confidence intervals are found using the t-distribution (especially if you are working with small samples). Watch the video for an example:


Can’t see the video? Click here.

Contents (Click to Skip to Section)

How to Find a Confidence Interval by Hand:

  1. How to Find a Confidence Interval for a Sample (T-Distribution)
  2. How to Find a Confidence Interval for a Sample (Example 2)
  3. How to Find a Confidence Interval with the Normal Distribution / Z-Distribution
  4. How to Find a Confidence Interval for a Proportion
  5. How to Find a Confidence Interval for Two Populations (Proportions)

How to Find a Confidence Interval using Technology:

  1. Confidence Interval for the Mean in Excel
  2. Confidence Interval on the TI 83: Two Populations;
  3. Using the TI 83 to Find a Confidence Interval for Population Proportion, p
  4. TI 83 Confidence Interval for the Population Mean
  5. Confidence Interval for a Mean on the TI 89
  6. Confidence Interval for a Proportion on the TI 89

Explanations and Definitions:

  1. The 95% Confidence Interval Explained
  2. Asymmetric Confidence Interval
  3. Calculating Confidence Intervals (Part III of Intro to Statistics)

See also:

Binomial Confidence Intervals.
What is a Z Interval?

What is the Definition of a Confidence Interval?

A confidence interval is how much uncertainty there is with any particular statistic. Confidence intervals are often used with a margin of error. It tells you how confident you can be that the results from a poll or survey reflect what you would expect to find if it were possible to survey the entire population. Confidence intervals are intrinsically connected to confidence levels.

Confidence Intervals vs. Confidence Levels

Confidence levels are expressed as a percentage (for example, a 95% confidence level). It means that should you repeat an experiment or survey over and over again, 95 percent of the time your results will match the results you get from a population (in other words, your statistics would be sound!). Confidence intervals are your results and they are usually numbers. For example, you survey a group of pet owners to see how many cans of dog food they purchase a year. You test your statistic at the 99 percent confidence level and get a confidence interval of (200,300). That means you think they buy between 200 and 300 cans a year. You’re super confident (99% is a very high level!) that your results are sound, statistically.

confidence intervals

Image: WUSTL.EDU

Real Life Examples of Confidence Intervals

A 2008 Gallup survey found that TV ownership may be good for wellbeing. The results from the poll stated that the confidence level was 95% +/-3, which means that if Gallup repeated the poll over and over, using the same techniques, 95% of the time the results would fall within the published results. The 95% is the confidence level and the +/-3 is called a margin of error. At the beginning of the article you’ll see statistics (and bar graphs). At the bottom of the article you’ll see the confidence intervals. For example, “For the European data, one can say with 95% confidence that the true population for wellbeing among those without TVs is between 4.88 and 5.26.” The confidence interval here is “between 4.88 and 5.26“.

The U.S. Census Bureau routinely uses confidence levels of 90% in their surveys. One survey of the number of people in poverty in 1995 stated a confidence level of 90% for the statistics “The number of people in poverty in the United States is 35,534,124 to 37,315,094.” That means if the Census Bureau repeated the survey using the same techniques, 90 percent of the time the results would fall between 35,534,124 and 37,315,094 people in poverty. The stated figure (35,534,124 to 37,315,094) is the confidence interval.
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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.
how to find a 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
tα=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.

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How to Find a Confidence Interval Example 2 (small sample)

Watch the video for an example:


Can’t see the video? Click here.

If you have one small set of data (under 30 items), you’ll want to use the t-distribution instead of the normal distribution to construct your confidence interval.

construct a confidence interval

The formula for constructing a CI with the t-distribution.

Example problem: Construct a 98% Confidence Interval based on the following data: 45, 55, 67, 45, 68, 79, 98, 87, 84, 82.

Step 1: Find the mean, μ and standard deviation, σ for the data.
σ: 18.172.
μ: 71
Set these numbers aside for a moment.

Step 2: Subtract 1 from your sample size to find the degrees of freedom (df). We have 10 numbers listed, so our sample size is 10, so our df = 9. Set this number aside for a moment.

Step 3: Subtract the confidence level from 1, then divide by two. This is your alpha level.
(1 – .98) / 2 = .01

Step 4: Look up df (Step 2) and α (Step 3) in the t-distribution table. For df = 9 and α = .01, the table gives us 2.821.

degrees of freedom

Degrees of freedom in the left column of the t distribution table.



Step 5: Divide your std dev (step 1) by the square root of your sample size.
18.172 / √(10) = 5.75

Step 6: : Multiply step 4 by step 5.
2.821 × 5.75 = 16.22075

Step 7: For the lower end of the range , subtract step 6 from the mean (Step 1).
71 – 16.22075 = 54.77925

Step 8: For the upper end of the range, add step 6 to the mean (Step 1).
71 + 16.22075 = 87.22075

That’s how to find a confidence interval using the t-distribution!
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Confidence Interval with the Normal Distribution / Z-Distribution

Watch the video for an example:


Can’t see the video? Click here.

If you don’t know your population mean (μ) but you do know the standard deviation (σ), you can find a confidence interval for the population mean, with the formula:
x̄ ± z* σ / (√n),

Example problem: Construct a 95 % confidence interval an experiment that found the sample mean temperature for a certain city in August was 101.82, with a population standard deviation of 1.2. There were 6 samples in this experiment.

Step 1: Subtract the confidence level (Given as 95 percent in the question) from 1 and then divide the result by two. This is your alpha level, which represents the area in one tail.
(1 – .95) / 2 = .025

Step 2: Subtract your result from Step 1 from 1 and then look that area up in the middle of the z-table to get the z-score:

  1. 1 – 0.025 = 0.975
  2. z score = 1.96.

Step 3: Plug the numbers into the second part of the formula and solve:
z* σ / (√n)
= 1.96 * 1.2/√(6)
= 1.96 * 0.49
= 0.96

Step 4: For the lower end of the range, subtract step 3 from the mean.
101.82 – 0.96 = 100.86

Step 5: For the upper end of the range, add step 3 to the  mean.
101.82 + 0.96 = 102.78.

The CI is (100.86,102.78)

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How to Find a Confidence Interval for a Proportion: Overview

Watch the video for an example of finding a confidence interval for population proportion of successes (and failures):


Can’t see the video? Click here.

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

Watch the video for another example:


Can’t see the video? Click here.

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:

  • Find z α/2. You don’t have to look this up in the z-table every time, you can find common ones in this table:
    z alpha/2

    According to the table, for a 90% CI, z α/2 = 1.645.
  • p-hat: Divide the proportion given (i.e. the smaller number)by the sample size. 57/510 = 0.112
  • q-hat: To find q-hat, subtract p-hat (from directly above) from 1. This gives: 1 – 0.112 = 0.888

Step 2: Multiply p-hat by q-hat (from Step 1).
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 1.645 = 0.023.

Step 6:: For the lower percentage, subtract step 5 from p-hat.  
0.112 – 0.023 =  0.089 = 8.9%.

Step 7:For the upper percentage, add step 5 to p-hat.
0.112 + 0.023 =  13.5%.

This next method involves plugging in numbers into the actual formula. You’ll get the same results if you use the “formula free” method above or if you use the steps below.

Confidence intervals for a proportion are calculated using the following formula:
confidence interval for a proportion 1
The formula might look daunting, but all you really need are two pieces of information: the z-score and the P-hat. You should be familiar with looking up z-scores from previous sections on the normal distribution (if you need a refresher, be sure to watch the above video) and P-hat is just dividing the number of events by the number of trials. Once you’ve figured those two items out, the rest is basic math.

Confidence Interval for a Proportion Example 2: Steps

Example question: Calculate a 95% confidence interval for the true population proportion using the following data:
Number of trials(n) = 160
Number of events (x) = 24

Step 1: Divide your confidence level by 2: .95/2 = 0.475.

Step 2: Look up the value you calculated in Step 1 in the z-table and find the corresponding z-value. The z-value that has an area of .475 is 1.96.
confidence interval for a proportion 2

Step 3: Divide the number of events by the number of trials to get the “P-hat” value: 24/160 = 0.15.

Step 4: Plug your numbers into the formula and solve:

  1. 0.15 ± (1.96) √ ((0.15(1 – 0.15) / 160))=
  2. 0.15 ± (1.96) √ ((0.15(0.85)/160))=
  3. 0.15 ± (1.96) √ ((0.1275)/160))=
  4. 0.15 ± (1.96) √ (0.000796875)=
  5. 0.15 ± (1.96) 0.0282289744765905=
  6. 0.15 ± 0.0553 =
  7. 0.15 – 0.0553 = 0.0947 <--this is your lower confidence interval for a proportion
  8. 0.15 + 0.0553 = 0.2053 <--this is your upper confidence interval for a proportion

Your answer can be expressed as: (0.0947,0.2.053).
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How to Find a Confidence Interval for Two Populations (Proportions)

Finding confidence intervals for two populations can look daunting, especially when you take a look at the ugly equation below.
confidence intervals for two populations
It looks a lot worse than it is, because the right side of the equation is actually a repeat of the left! Finding confidence intervals for two populations can be broken down to an easy three steps.

Example question: A study revealed that 65% of men surveyed supported the war in Afghanistan and 33% of women supported the war. If 100 men and 75 women were surveyed, find the 90% confidence interval for the data’s true difference in proportions.

Step 1: Find the following variables from the information given in the question:
n1 (population 1)=100

Phat1 (population 1, positive response): 65% or 0.65
Qhat1 (population 1, negative response): 35% or 0.35
n2(population 2)=75
Phat2 (population 2, positive response): 33% or 0.33
Qhat2 (population 2, negative response): 67% or 0.67

Step 2: Find zα/2
(If you’ve forgotten how to find α/2, see the directions in: How to Find a Confidence Interval for a Proportion above)
zα/2=0.13

Step 3: Enter your data into the following formula and solve:
two populations CI proportion

If formulas scare you, here’s the step-by-step to solve the equation (refer back to step 1 for the variables):

  1. multiply phat1 and qhat1 together (.65 x .35 = .2275)
  2. divide your answer to (1) by n1. Set this number aside. (.2275 x 100=.00275)
  3. multiply phat2 and qhat2 together (.33 x .67=.2211).
  4. divide your answer to (3) by n2 (.2211/75=.002948).
  5. Add (3) and (4) together (.00275 + .002948=.005698)
  6. Take the square root of (5): (sqrt.005698=.075485)
  7. Multiply (6) by zα/2 found in Step 2. (.075485 x 0.13=.0098). Set this number aside.
  8. Subtract phat2 from phat1 (.65-.33=.32).
  9. Subtract (8) from (7) to get the left limit (.32-0.0098 = 31.9902)
  10. Add (7) to (8) to get the right limit (.32+.0.0098=32.0098)

That’s it!
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Confidence Interval for the Mean in Excel

Watch the video for an example:


Can’t see the video? Click here.

How to Find a Confidence Interval for the Mean in Excel: Overview

confidence interval for the mean formula

CI for the mean formula.

A confidence interval for the mean is a way of estimating the true population mean. Instead of a single number for the mean, a confidence interval gives you a lower estimate and an upper estimate. For example, instead of “6” as the mean you might get {5,7}, where 5 is the lower estimate and 7 is the upper. The narrower the estimate, the more precise your estimate is. The equations involved in statistics often involve a lot of minor calculations (such as summation), plus you would also need to calculate the margin of error and the mean of the sample. It’s very easy for errors to slip in if you calculate the confidence interval by hand. However, Excel can calculate the mean of the sample, the margin or error and confidence interval for the mean for you. All you have to do is provide the data —which for this technique must be a sample greater than about 30 to give an accurate confidence interval for the mean.

How to Find a Confidence Interval for the Mean in Excel: Steps

Example problem:Calculate the 95 percent confidence interval for the mean in Excel using the following sample data: 2, 5, 78, 45, 69, 100, 34, 486, 34, 36, 85, 37, 37, 84, 94, 100, 567, 436, 374, 373, 664, 45, 68, 35, 56, 67, 87, 101, 356, 56, 31.
Step 1: Type your data into a single column in Excel. For this example, type the data into cells A1:A31.

Step 2: Click the “Data” tab, then click “Data Analysis,” then click “Descriptive Statistics” and “OK.” If you don’t see Data Analysis, load the Excel data analysis toolpak.
confidence interval for the mean excel

Step 3: Enter your input range into the Input Range box. For this example, your input range is “A1:A31”.

Step 4: Type an output range into the Output Range box. This is where you want your answer to appear. For example, type “B1.”

Step 5: Click the “Summary Statistics” check box and then place your chosen confidence level into the ‘Confidence Level for Mean’ check box. For this example, type “95”.
ci mean excel

Step 5: Click “OK.”Microsoft Excel will return the confidence interval for the mean and the margin of error for your data. For this sample, the mean (Xbar) is 149.742 and the margin of error is 66.9367. So the mean has a lower limit of 149.742-66.936 and an upper limit of 149.742+66.936.
confidence interval for the mean excel results
That’s it!

Warning: A 99 percent confidence interval doesn’t mean that there’s a 99 percent probability that the calculated interval has the actual mean. Your sample is either going to contain the actual mean, or it isn’t. Over the long-term, if you ran tests on many, many samples, there is a 99 percent probability that the calculated intervals would contain the true mean.
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TI 83 Confidence Interval:Two Populations

Confidence Interval TI 83
Statistics about two populations is incredibly important for a variety of research areas. For example, if there’s a new drug being tested for diabetes, researchers might be interested in comparing the mean blood glucose level of the new drug takers versus the mean blood glucose level of a control group. The confidence interval(CI) for the difference between the two population means is used to assist researchers in questions such as these.

The TI 83 allows you to find a CI for the difference between two means in a matter of a few keystrokes.

Example problem: Find a 98% CI for the difference in means for two normally distributed populations with the following characteristics:

x1 = 88.5
σ1 = 15.8
n1 = 38

x2 = 74.5
σ2= 12.3
n2 = 48

Step 1: Press STAT, then use the right arrow key to highlight TESTS.
ti calculator confidence

Step 2: Press 9 to select 2-SampZInt….

Step 3: Right arrow to Stats and then press ENTER. Enter the values from the problem into the appropriate rows, using the down arrow to switch between rows as you complete them.
ti calculator steps

Step 4: Use the down arrow to select Calculate then press ENTER.
ti calculator confidence intervals

The answer displayed is (6.7467, 21.253). We’re 98% sure that the difference between the two means is between 6.7467 and 21.253.

That’s it!

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How to Find a Confidence Interval for Population Proportion, p on the TI 83

Watch the video for the steps:


Can’t see the video? Click here.

Example problem: A recent poll shows that 879 of 1412 Americans have had at least one caffeinated beverage in the last week. Construct a 90% confidence interval for p, the true population proportion.

Note:

  • “x” is the number of successes and must be a whole number. Successes in this question is how many Americans have had at least one caffeinated beverage (879). If you are given p̂ instead (the sample proportion), multiply p̂ by n to get x (because x=n* p̂).
  • “n” is the number of trials.

Step 1: Press STAT.

Step 2: Right arrow over to “TESTS.”

Step 3: Arrow down to “A:1–PropZInt…” and then press ENTER.

Step 4: Enter your x-value: 879.

Step 5: Arrow down and then enter your n value: 1412.

Step 6: Arrow down to “C-Level” and enter .90. This is your confidence level and must be entered as a decimal.

Step 7: Arrow down to calculate and press ENTER. The calculator will return the range (.6013, .64374)


That means the 98 percent CI for the population proportion is between 0.6013 and .64374.

Tip: Instead of arrowing down to select A:1–PropZInt…, press Alpha and MATH instead.
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How to Find a Confidence Interval on the TI 83 for the Population Mean

If you don’t know how to enter data into a list, you can find the information in this article on TI 83 cumulative frequency tables.

Watch the video for the steps:


Can’t see the video? Click here.

Example problem: 40 items are sampled from a normally distributed population with a sample mean x̄ of 22.1 and a population standard deviation(σ) of 12.8. Construct a 98% confidence interval for the true population mean.

Step 1: Press STAT, then right arrow over to “TESTS.”

Step 2: Press 7 for “Z Interval.”

Step 3: Arrow over to “Stats” on the “Inpt” line and press ENTER to highlight and move to the next line, σ.

Step 4: Enter 12.8, then arrow down to x̄.

Step 5: Enter 22.1, then arrow down to “n.”

Step 6: Enter 40, then arrow down to “C-Level.”

Step 7: Enter .98. Arrow down to “calculate” and then press ENTER. The calculator will give you the result of (17.392, 26.808) meaning that your 98% confidence interval is 17.392 to 26.808. This is the same as:

17.392 > μ > 26.808

That’s how to find a Confidence Interval on the TI 83 for the Population Mean!

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How to Find a Confidence Interval for the Mean on the TI 89

Example problem #1 (known standard deviation): Fifty students at a Florida college have the following grade point averages: 94.8, 84.1, 83.2, 74.0, 75.1, 76.2, 79.1, 80.1, 92.1, 74.2, 64.2, 41.8, 57.2, 59.1, 65.0, 75.1, 79.2, 95.0, 99.8, 89.1, 59.2, 64.0, 75.1, 78.2, 95.0, 97.8, 89.1, 65.2, 41.9, 55.2. Find the 95% confidence interval for the population mean, given that σ = 2.27.

Step 1: Press APPS and scroll to Stats/List Editor. ENTER.

Step 2: Press F1 then 8. This clears the list editor.

Step 3: Press ALPHA ) 9 2 to name the list “CI2.”

Step 4: Enter your data in a list. Follow each number with the ENTER key: 94.8, 84.1, 83.2, 74.0, 75.1, 76.2, 79.1, 80.1, 92.1, 74.2, 64.2, 41.8, 57.2, 59.1, 65.0, 75.1, 79.2, 95.0, 99.8, 89.1, 59.2, 64.0, 75.1, 78.2, 95.0, 97.8, 89.1, 65.2, 41.9, 55.2.

Step 5: Press F4 then 1.

Step 6: Enter “ci” in the “List” box: ALPHA key then ) 9 2.

Step 7: Enter 1 in the frequency box. Press ENTER. This should give you the mean (xbar, the first in the list) = 75.033.

Step 8: Press ENTER. Press 2nd F7 1 ENTER. This brings up the z-distribution menu.

Step 9: Press the right arrow key then the down arrow to select a “Data Input Method” of “Stats.” Press ENTER.

Step 10: Enter your σ from the question (in our case, 2.27), xbar from Step 7 (75.3033), n = 30 and the Confidence Interval from the question (in our example, it’s .95).

Step 11: Press ENTER and read the results. The “C Int” is {74.49,76.123}. This means we are 95% confident that the population mean falls between 74.49 and 76.123.

That’s it!

Example problem #2 (unknown standard deviation): A random sample of 30 students at a Florida college has the following grade point averages: 59.1, 65.0, 75.1, 79.2, 95.0, 99.8, 89.1, 65.2, 41.9, 55.2, 94.8, 84.1, 83.2, 74.0, 75.1, 76.2, 79.1, 80.1, 92.1, 74.2, 59.2, 64.0, 75.1, 78.2, 95.0, 97.8, 89.1, 64.2, 41.8, 57.2. What is the 90% confidence interval for the population mean?

Step 1: Press APPS. Scroll to the Stats/List Editor and press ENTER.

Step 2: Press F1 8 to clear the editor.

Step 3: Press ALPHA ) 9 to name the list “CI.”

Step 4: Enter your data in a list. Follow each number with the ENTER key: 59.1, 65.0, 75.1, 79.2, 95.0, 99.8, 89.1, 65.2, 41.9, 55.2, 94.8, 84.1, 83.2, 74.0, 75.1, 76.2, 79.1, 80.1, 92.1, 74.2, 59.2, 64.0, 75.1, 78.2, 95.0, 97.8, 89.1, 64.2, 41.8, 57.2.

Step 5: Press F4 1.

Step 6: Enter “ci” in the List box: Press ALPHA ) 9.

Step 7: Enter1 in the frequency box. Press ENTER. This should give you the sample standard deviation, sx = 15.6259, n = 30, and x (the sample mean) = 75.033.

Step 8: Press ENTER. Press 2nd F2 2.

Step 9: Press the right arrow key then the down arrow to select a “Data Input Method” of “Stats.” Press ENTER.

Step 10: Enter your x, sx and n from Step 7. In our example, sx = 15.6259. n = 30 and x = 75.033.

Enter the Confidence Interval from the question (in our example, it’s .9).

Step 11: Press ENTER and read the results. The C Int is {70.19,79.88} which means that we are 90% confident that the population mean falls between 70.19 and 79.88.

That’s it!

Tip: If you know σ, use ZInterval instead of TInterval.
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How to find a Confidence Interval for a Proportion on the TI 89

Example problem #1: In a simple random sample of 295 students, 59.4% of students agreed to a tuition increase to fund increased professor salaries. What is the 95% CI for the proportion in the entire student body who would agree?

Step 1: Press APPS and scroll down to Stats/List Editor. Press ENTER.
If you don’t see the Stats/List editor, download it HERE from the TI-website. You’ll need the graphlink cable that came with your calculator to transfer the software.

Step 2: Press 2nd F2 5 for the 1-PropZInt menu.

Step 3: Figure out your “successes.” Out of 295 people, 59.4% said yes, so .694 × 295 = 175 people.

Step 4: Enter your answer from Step 3 into the Successes,x box: 175.

Step 5: Scroll down to n. Enter 295, the number in the sample.

Step 6: Scroll down to C Level. Enter the given confidence level. In our example, that’s .95. Press ENTER twice.

Step 7: Read the result. The calculator returns the result C Int {.5372, .6493}. This means that you are 95% confident that between 54% and 65% of the student body agree with your decision.

Tip: If you are asked for a folder when entering the Stats Editor, just press Enter. It doesn’t matter which folder you use.

Warning: Make sure your round your “success” entries to the nearest integer to avoid a domain error.

Example problem #2: A recent poll in a simple random sample of 986 women college students found that 699 agreed that textbooks were too expensive. Out of 921 men surveyed by the same manner, 750 thought that textbooks were too expensive. What is the 95% confidence interval for the difference in proportions between the two populations?

Step 1: Press APPS, scroll to the Stats/List Editor, and press ENTER.

Step 2: Press 2nd F2 6 to reach 2-PropZint.

Step 3: Enter your values into the following boxes (Use “women” for population 1 (x1 and n1) and “men” for population 2 (x2 and n2)):

  • Successes, x1: 590*
  • n1: 796
  • Successes, x2: 548
  • n2: 800
  • C Level: 0.95

Step 4: Press ENTER.

Step 5: Read the result. The confidence interval is displayed at the top as C Int { .0119,.10053}. This means that your confidence interval is between 1.19% and 10.05%.

That’s how to find a Confidence Interval on the TI 89!

Tip: As long as you keep track of which population is x1/n1 and x2/n2, it doesn’t matter which is entered in which box.

*You must enter a whole number here, or you’ll get ERR:DOMAIN.

You’ll come across this common type of problem in elementary stats: find a confidence interval given a large random sample and the number of “successes” in that sample.

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The 95% Confidence Interval Explained

The terms confidence level and confidence interval are often confused.

A 95% confidence level means is that if the survey or experiment were repeated, 95 percent of the time the data would match the results from the entire population. Sometimes you just can’t survey everyone because of time or cost (think about how much it would cost to do a telephone survey of over 300 million Americans!).Therefore, you take a sample of the population. Having a 95% confidence level means that you’re almost certain your results are the same as if you had surveyed everyone.

A 95% confidence interval gives you a very specific set of numbers for your confidence level. For example, let’s suppose you were surveying a local school to see what the student’s state test scores are. You set a 95% confidence level and find that the 95% confidence interval is (780,900). That means if you repeated this over and over, 95 percent of the time the scores would fall somewhere between 780 and 900.

95% confidence interval

Image: WUSTL.EDU

The above image shows a 95% confidence interval on a normal distribution graph. The red “tails” are the remaining 5 percent of the interval. Each tail has 2.5 percent (that’s .025 as a decimal). You don’t have to draw a graph when you’re working with confidence intervals, but it can help you visualize exactly what you are doing — especially in hypothesis testing. If your results fall into the red region, then that’s outside of the 95% confidence level that you, as a researcher, set.

If you have a small sample or if you don’t know the population standard deviation which in most real-life cases is true), then you’ll find the 95% Confidence Interval with a t-distribution.

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Asymmetric Confidence Interval

An asymmetric confidence interval just means that the point estimate doesn’t lie in the exact center of the CI. You can end up with asymmetric CIs for many reasons, including:

  • You transform your data (for example, using log transformations).
  • You incorporate random error.
  • You incorporate systematic bias into the interval:
    • A positive systematic bias will increase the right side of the interval.
    • A negative systematic bias will increase the left side of the interval.

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Calculating Intervals

You may want to read Part 1 and 2 of Intro to Statistics before reading this section.

With random sampling of binomial values (in-favor vs. not-in-favor; heads vs. tails):

  1. Sampling from populations with percent-in-favor close to 50% have wider sampling distributions than populations with percentages closer to 0% or 100%.
  2. Larger sample sizes have narrower sampling distributions.

The various sampling distributions have different locations on the horizontal axis and they have different widths. It would be useful to convert them all to one standard scale. We’ll need a common unit. And the rescaling to that unit must account for the effects of the population percent-in-favor value (number 1above) and sample size (number 2 above).

The unit to be used is called Standard Error. It’s labeled “Standard” because it serves as a standard unit. And it’s labelled “Error” because we don’t expect our sample statistic values to be exactly equal to the population parameter; there will be some amount of error. The Standard Error formula, which I’ll explain a piece at a time, is as follows:
standard error formula

The variable p is the proportion rather than percentage: .5 rather than 50% (and 0 rather than 0%; .01 rather than 1%; .1 rather than 10%; and 1 rather than 100%).
The p * (1 – p) term in the numerator is called the proportion variance. Sampling from populations with percent-in-favor close to 50% have wider sampling distributions than populations with percentages closer to 0% or 100%.

The variance p * (1 – p) reflects this dynamic:

  • 0.0 * (1 – 0) = 0.00
  • .01 * (1 – .01) = .01
  • .1 *(1 – .1) = .09
  • .3 *(1 – .3) = .21
  • .5 *(1 – .5) = .25
  • .7 *(1 – .7) = .21
  • .9 *(1 – .9) = .09
  • .99 *(1 – .99) = .01
  • 1.0 *(1 – 1) = 0.00

So, as p moves from .5 towards 0 or 1, variance decreases, and since variance is in the numerator, Standard Error decreases. Decreases in Standard Error correspond to narrowing of the sampling distribution. This reflects lower uncertainty. Lower variance, lower uncertainty. Variance is itself a statistic and is very important in statistical analysis. We’ll be seeing it in formulas from now on. Now let’s consider sample size, which is represented in the denominator of the formula by n.

Larger sample sizes have narrower sampling distributions. Since n is in the denominator of the Standard Error formula, as n increases Standard Error decreases. Again, decreases in Standard Error correspond to narrowing of the sampling distribution. Again, this reflects lower uncertainty. Larger sample size, lower uncertainty.

Now we can use the Standard Error scale to determine 95% intervals. First, an important fact: The boundary lines of the 95% interval on the Standard Error scale are always -2 and +2 (they’re actually -1.95996… and +1.95996…, but I’m rounding to -2 and +2 for the present purposes). Let’s clarify all this by looking at several example calculations and illustrations.

Let’s start with random sampling of 100 from a population that is 50% in favor of the new public health policy (Figure 1.2, below).
intro to statistics figure 2

Plugging in the numbers gives
solved formula for random sampling

Standard Error is .05 and two Standard Errors is .1 in proportions and 10% in percentages. Since we want to center the interval on the percentage p of 50%, we’ll add and subtract 10% from 50%. This yields a calculated 95% interval of 50% + 10% (50% minus 10% to 50% plus 10%) or 40%-to-60%. That’s also what Figure 1.2 shows!

Putting everything we just computed into a formula for calculating 95% intervals we get
formula for calculating 95% interval

Next let’s consider the 95% interval of random sampling of 100 from a population that is 30% in favor of the new public health policy (Figure 2.7, reproduced below).
95% interval of random sampling

Standard Error is .045 and two Standard Errors is .09 in proportions and 9% in percentages. We want to center the interval on 30%, so we’ll add and subtract 9% from 30%. This yields a 95% interval of 30% ± 9% (30% minus 9% to 30% plus 9%) or 21%-to-39%. That’s also what Figure 2.7 shows!
standard errors

Last let’s consider the 95% interval of random sampling of 1000 from a population that is 50% in favor of the new public health policy (Figure 2.3, below).
solved standard error formula

Standard Error is .015 and two Standard Errors is .03 in proportions and 3% in percentages. We want to center the interval on 50%, so we’ll add and subtract 3% from 50%. This yields a 95% interval of 50% + 3% or 47%-to-53%. That’s also what Figure 2.3 shows!
95% confidence interval solved

The formula works! The reason the formula works is because the sampling distributions are “bell shaped”. More than that, they approximate the very special bell shape called the Normal distribution.

Let’s go one step further and standardize an entire sampling distribution to get what’s called the Standard Normal distribution. The Standard Normal Distribution is a normal distribution that uses Standard Error as its unit (rather than percentages or proportions). To illustrate, let’s standardize Figure 1.1 (below).
intro to statistics

Figure 3.1 is a standardized version of Figure 1.1.
standardized standard error

Notice that Standard Error is the unit used on the horizontal axis of Figure 3.1. This is done by rescaling the horizontal axis unit of Figure 1.1 to the Standard Error unit of Figure 3.1 using the below formula.
rescaling the horizontal axis

This formula gives us how many Standard Errors a proportion, p, is from .5. First, we convert the percentages to proportions. Next, we recenter the axis: whereas Figure 1.1 is centered on the proportion value .5 (50%), Figure 3.1 is centered on zero Standard Errors; the numerator p-.5 centers the horizonal axis of Figure 3.1 onto zero. Finally, these differences are divided by the Standard Error to rescale the horizontal axis. Voila, Figure 1.1 has been standardized to the Standard Error scale of Figure 3.1.

Figure 3.2 shows its 95% interval below Figure 1.2.
intro to stats 95% interval

95% interval

The boundary lines of the 95% interval on the Standard Error scale are -2 and 2 (rounded). Plugging .4 (40%) and .6 (60%) from Figure 1.2 into the above formula gives us -2 and 2 Standard Errors as the 95% boundary lines in the Standard Error unit. As emphasized above: The boundary lines of the 95% interval on the Standard Error scale are always -2 and +2 (rounded). If we standardized Figures 2.3 and 2.7,
we’ll again find the 95% interval boundary lines to be -2 and 2. (You can use the formula and do the arithmetic if you want to confirm this.)

We can convert our units (e.g., percent-in-favor, percent-heads) into the Standard Error unit and vice versa by multiplying and dividing by Standard Error. That comes in very handy. All of the sampling distributions we’ve looked at so far can be standardized in this way. In practice, we don’t convert entire sampling distributions to the standardized distribution; we use Standard Error in formulas as multipliers and divisors to calculate individual values, like we do to calculate the boundary lines for 95% intervals and to convert proportions to the Standard Error scale.

convert proportions to standard errors

A Survey Example

We’ll further explore the Standard Normal distribution later on, but first let’s put some of what we’ve covered so far into action, while also expanding our horizons. 1,000 Surveyors’ Sample Statistics and Their 1,000 95% Confidence Intervals. In this section we’re going to look at things from a different perspective. Surveyors won’t be comparing their sample statistics of public opinion with what to expect when the population opinion statistic equals a particular value, like 50%. Instead, the surveyors want to determine, based on their sample statistic, what the value of
the population statistic might be. For example, a surveyor who gets a sample statistic value of 34% will want to calculate a 95% interval surrounding 34% and explain what that interval might tell us about the overall population’s opinions.

We are going to explore the subtleties involved by sending out 1,000 surveyors to survey the same population and see what they come up with and how they should interpret what they come up with. But first we’ll need to set the stage by inventing a population that has certain characteristics that we know, but none of the surveyors know. Our invented community, Artesian Wells, has about 70,000 residents. There is a
new public health policy being debated and, since we are all-knowing, we know that 40% of the residents agree with the new policy. Only we know this. We want to know what to expect when many, many surveyors randomly sample 1000 people from this population. The survey respondents will be asked whether they “agree” or “disagree” with something, a binomial response. We’ll use proportions rather than percentages, with the proportions rounded to two decimal places. Figure 3.3 shows us the sampling distribution of what to expect. (Don’t get confused: There are 1,000 random samples, and each sample has a sample size of 1000.)
a survey example

Based on visual inspection, notice that the great majority of the sample proportions are in the interval 0.37 to 0.43. Approximately 950 of the 1,000 sample proportions are contained within the interval 0.37 to 0.43, indicating that 0.37 to 0.43 is the 95% interval surrounding the population proportion of 0.40. The formula will give us the same boundary lines. (Feel free to double check.)

As always, we expect the 95% interval around the population proportion to contain 95% of all sample proportions obtained by random sampling.
Now, we hire 1,000 independent surveyors who converge on the town to do the “agree” or “disagree” survey. All 1,000 surveyors get their own random sample of 1000 residents and calculate their own sample proportion-agree statistic. How does each of the individual surveyors analyze their sample proportion?

First, let’s look at the formula for calculating 95% confidence intervals for sample proportions. It looks much like the formula in the previous section. The variable p with a hat on denotes the sample proportion (as opposed to the population proportion). The square root term calculates the Standard Error for the sample proportion. Sample size is again represented by n. As for the constant 1.96, recall that earlier I rounded +1.95996…Standard Errors and used +2 Standard Errors; now I’m being more precise by using +1.96 Standard Errors, which is more common.
+1.96 Standard Errors

Each of the 1,000 surveyors calculates their individual interval using their sample proportion value, and we expect that 95% of the surveyors’ 95% confidence intervals will contain the population proportion (0.4 in this example). You might want to reread that sentence a few times, keeping in mind that although we, as know-it-alls, know that the population proportion is 0.4, none of the surveyors have any idea what it is.

It’s in this context that the term “confidence” in “confidence interval” came about: we are confident that 95% of all 95% confidence intervals for sample statistic values obtained via random sampling will contain the population statistic value. (But no individual surveyor will know whether their confidence interval contains the population statistic value or not!)

In a nutshell:

  • The 1,000 surveyors calculate their individual 95% confidence intervals.
  • About 950 of them will have an interval containing the population proportion.
  • About 50 of them won’t.

Let’s look at the 95% confidence intervals constructed via the formula by two surveyors: The first got a sample proportion of 0.38, and the second got a sample proportion of 0.34.
Surveyor #1 Result.
Using the formula with a sample proportion of 0.38 and a sample size of 1000, the 95% confidence interval is 0.35 to 0.41 (rounded).
Only we, being know-it-alls, know that this 95% confidence interval contains the population proportion of 0.4
Surveyor #2 Result.
Using the formula with a sample proportion of 0.34 and a sample size of 1000, the 95% confidence interval is 0.31 to 0.37. Only we, being know-it-alls, know that this 95% confidence interval does not contain the population proportion of 0.4. Only we know that this surveyor is one of the 5% of unlucky surveyors who just happened to get a misleading random sample. This is called a Type I Error.

In summary,

  1. We expect the 95% confidence interval around the population proportion to contain 95% of all sample proportions obtained by random sampling. We’ve been seeing this all along.
  2. We expect 95% of all the 95% confidence intervals based on random sample proportions to contain the population proportion. We see this for the first time here; more detail is given next.

Table 3.1 is divided into three sections, left to right, and shows what the various surveyors will get. Overall, the Table shows the confidence intervals for surveyors with sample proportions of 0.3 through 0.5; sample proportions 0.30 through 0.36 are in the left section, 0.37 through 0.43 are in the middle section (shaded), and 0.44 through 0.50 are in the right section. Notice that the expected 950 surveyors in the middle section (shaded) with sample proportions within the interval of 0.37 to 0.43 also have 95% confidence intervals that contain the population proportion
of 0.40. The expected 50 surveyors with sample proportions outside the interval of 0.37 to 0.43—the left and right sections of the Table—do not have 95% confidence intervals that contain the population proportion of 0.40.
95% confidence intervals that contain the population proportion

Again, in summary, and for emphasis:

  1. We expect the 95% confidence interval around the population proportion to contain 95% of all sample proportions obtained by random sampling.
  2. We expect 95% of all the 95% confidence intervals based on random sample proportions to contain the population proportion.

Because of these two facts, we will reach the same conclusion whether we

  1. check if a sample proportion is outside the 95% interval surrounding a hypothesized population proportion, or
  2. check if the hypothesized population proportion is outside the 95% interval surrounding a sample proportion.

The analysis can be done either way.

Here’s a quick analogy: Suppose a stamping plant that makes coins was malfunctioning and produced unbalanced (i.e., unfair) coins. Unbeknownst to anyone, these unfair coins favored tails, and the chance of coming up heads is only 0.4. Now say 1,000 people flip these coins 1000 times each, while counting and then determining the proportion of heads. What would the 1,000 peoples’ results be like? What would an analysis of a single coin and its 1000 flips be like? Answer: Just like the survey example above. Just replace the words “agree” and “disagree” with “heads” and “tails”. We expect 95% of the coin-flippers will get 95% confidence intervals that contain 0.4, and 5% of the coin-flippers will get 95% confidence interval that do not contain 0.4. In other words, we expect 95% of the results to be veridical and 5% of the results to be misleading. But no one knows whether their results are veridical or misleading. The reason I use the word “veridical” is because it’s the perfect word: “Coinciding
with reality.” I’m using it to mean the opposite of misleading.

Next: Type I and Type II Errors

References

J.E. Kotteman. Statistical Analysis Illustrated – Foundations . Published via Copyleft. You are free to copy and distribute this content.

Kenney, J. F. and Keeping, E. S. “Confidence Limits for the Binomial Parameter” and “Confidence Interval Charts.” §11.4 and 11.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 167-169, 1962.

CITE THIS AS:
Stephanie Glen. "Confidence Interval: How to Find it: The Easy Way!" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/probability-and-statistics/confidence-interval/
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