Finding Confidence Intervals for Two Populations (Proportions)

Confidence Intervals 7 Comments

Finding confidence intervals for two populations can look daunting, especially when you take a look at the ugly equation below. It’s a lot worse that it looks, because the right side of the equation is actually a repeat of the left!). Finding confidence intervals for two populations can be done in an easy three steps:

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, refer this article on confidence levels.)
zα/2=1.65

Step 3: Enter your data into the following formula and solve:
twopopulationsCIproportion

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 (2) 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 1.65=.12455). Set this number aside.
  8. Subtract phat2 from phat1 (.65-.33=.32).
  9. Subtract (7) from (8) to get the left limit (.32-.12455=.19545)
  10. Add (7) to (8) to get the right limit (.32+.12455=.44455)

That’s it!

Related posts:

  1. How to Use a TI-89 for Figuring out Confidence Intervals for Two Populations: (Proportions)
  2. How to Find a Two Populations Confidence Interval With a TI 83 Calculator
  3. How to Use a TI-83 for Figuring out Confidence Intervals for The Population Mean (Known Standard Deviation)
  4. How to Use a TI-83 to Find a Confidence Interval for the Population Proportion, p
  5. How to Figure Out Confidence Intervals for a Proportion Using the TI-89 Calculator
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7 Responses to “Finding Confidence Intervals for Two Populations (Proportions)”

  1. Lisa Barcomb said:

    Nov 01, 09 at 8:10 pm

    This problem dealing with confidence intervals was the most grooling problem that I think I have come across. I just could not get it and I still don’t understand it. I feel somewhere along the line that this kind of problem doesn’t even exist in the real world. I know I am not a genius but get real this kind of information is out there in space somewhere.

  2. Bill Bryan said:

    Nov 02, 09 at 7:24 am

    It is very intimidating at first; the step-by-step run down is very helpful. I’m finding out that I struggle at first, but as I work thought the problem the end is very gratifying.

  3. mcguel said:

    Nov 02, 09 at 5:14 pm

    Professor Sundberg,

    I will work through this word problem, following the step by step guide.

    Thank you so much for assisting me; this is a difficult course, but with your help, I am learning the importance of statistics in every day life.

    Thanks again,

    Evelyn :-)

  4. Bill Bryan said:

    Nov 05, 09 at 1:47 pm

    So, if the very number is less than (<) a=____. Then reject, ***meaning there is enough evidence.

  5. Stephanie said:

    Nov 05, 09 at 2:20 pm

    Bill,
    You’d want to take additional steps before deciding to accept or reject (like finding the test value, for example). You can’t reject on just a confidence level alone,
    Stephanie

  6. angie widdows said:

    Nov 11, 09 at 6:00 pm

    Oh wow. This problem is a bear. I think this example helps because it explains line by line on what to do. thank you for this good example

  7. Special Force Guy said:

    Aug 07, 10 at 8:23 am

    Guys why don’t you place an online calculator on your website for the means of 2 populations ;(

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