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Rule of Sample Proportions and Confidence Intervals, Study notes of Elementary Mathematics

University of North Dakota (UND)Elementary Mathematics

The Rule of Sample Proportions (RoSP) and how to use it to construct confidence intervals for estimating the true population proportion when the true proportion is unknown. It includes the concept of sampling distribution, the 68-95-99.7 rule, and the process of building a confidence interval around a sample proportion.

What you will learn

  • How does the RoSP help in estimating the true population proportion?
  • What is a sampling distribution and how is it related to the RoSP?
  • What is the Rule of Sample Proportions?
  • How to construct a confidence interval around a sample proportion?
  • What is the difference between a 95% and a 90% confidence interval?

Typology: Study notes

2021/2022

Uploaded on 09/27/2022

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Rule of sample proportions
IF: 1. There is a population proportion of interest
2. We have a random sample from the population
3. The sample is large enough so that we will see at least five
of both possible outcomes
THEN: If numerous samples of the same size are taken and the sample
proportion is computed every time, the resulting histogram
will:
1. be roughly bell-shaped
2. have mean equal to the true population proportion
3. have standard deviation equal to:
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Download Rule of Sample Proportions and Confidence Intervals and more Study notes Elementary Mathematics in PDF only on Docsity!

Rule of sample proportions

IF: 1. There is a population proportion of interest

  1. We have a random sample from the population
  2. The sample is large enough so that we will see at least five of both possible outcomes THEN: (^) If numerous samples of the same size are taken and the sample proportion is computed every time, the resulting histogram will:
  3. be roughly bell-shaped
  4. have mean equal to the true population proportion
  5. have standard deviation equal to:

Sample proportions:

Suppose the true proportion is known

Recall that in craps, P(win) = 244/495 = 0.493 for each game.

Q: If you play 100 games of craps, what

proportion will you win?

A: Dunno.

Q: Okay, I guess that was obvious. But can we at

least give a range of possible proportions that will be

valid most (say, 95%) of the time?

A: Sure. Use the Rule of Sample Proportions.

True proportion unknown

Next, suppose we do not know the true population proportion value. This is far more common in reality! How can we use information from the sample to estimate the true population proportion? Suppose we have a sample of 200 students in STAT 100 and find that 28 of them are left handed. Our sample proportion is: 0.

We can now estimate the standard deviation of the sample proportion based on a sample of size 200: Hence, 2 standard deviations = 2×.025 =. On the following two slides, we’ll pretend that the true population proportion is 0.12.

If we repeat the

sampling over

and over, 95% of

our confidence

intervals will

contain the true

proportion of

This is why we

use the term

“95% confidence

interval”.

Definition of “95% confidence interval for the true population proportion” : An interval of values computed from the sample that is almost certain (95% certain in this case) to cover the true but unknown population proportion. The plan:

  1. Take a sample
  2. Compute the sample proportion
  3. Compute the estimate of the standard deviation of the sample proportion:
  4. 95% confidence interval for the true population proportion: sample proportion ± 2 ×SD

Suppose we want a 90% confidence interval instead of 95%. How many standard deviations span the middle 90% of the normal curve?

Other confidence coefficients:

An example

90% confidence interval

90% confidence interval: sample proportion ± 1.64×stdev

Since 90% is in the middle, there is 5% in either end. So find z for .05 and z for .95.

We get z = ±1.