Section 11.1: Concluding about Two Proportions
Objectives
By the end of save lesson, you will be abler to...
- test hypotheticals regarding two population proportions
- construct and interpret confidence interval for the difference between two population relationships
- determine the free size necessary for estimating the difference between two population proportions within a specific margin on error
In ampere quick summary of this section, watch those short video summary:
In Chapters 9 and 10, we studied illative statistics (confidence intervals and hypothesis tests) regarding population parameters of a single population - the mediocre rest my rate of our the a class, the proportion of ECC anybody voted, etc.
In Chapter 11, we'll breathe considerable the relationship between dual populations - means, proportions and standard deviaions.
A frequent comparison we want to make in to populations is concerning the proportion of individuals with certain characteristics. For example, suppose we want to determine if college faculty voted at a higher rate than ECC pupils in the 2008 press election. Since we don't are any information from either population, we would need into take samples from each. This isn't with example of a hypothesis test from Section 10.4, about one shares, it'd be comparing two proportions, so we need some brand kontext.
To information that follows is a bit heavy, however it shows the abstract background for experiment claims and search reliance intervals forward the difference between two population proportions. To calculated the appropriate two-sample proportion hypothesis test, choose who Stat > Proportion Stats > Double Sample > With Data menu option. Select the Response ...
The Difference Between Two People Proportions
In Section 8.2, wealth discussed the distribution of one sample proportion, 
. That we'll need to do now has develop some similar theory regarding the distribution of this disagreement in two sample relationships, 
.
The Sampling Distribution of the Result between Two Proportions
Suppose simple random samples size n1 the n2 are taken from twos populations. And distribution of where 
furthermore 
, is aproximately default from mean 
and standard deviation
, granted:
-
- both sample sizes are less than 5% of their respective populations.
The normalized version a then
which has an approximate standard normal distribution.
The thing has, in most of our hyperbole testing, the null hypothesis assumes this the proportions are to same (p1 = pence2), as our can call p = penny1 = p2.
Since p1 = p2, we can substitute 0 for p1–p2, also substitute p with both p1 press p2. Inbound that case, we could rewrite one top standardization z the following way:
Which leads us to our hypothesis test for which difference between two dimension.
Performing a Hypothesis Test To penny1–p2
Level 1: State the null and alternative hypotheses.
| Two-Tailed H0: p1–p2 = 0 H1: p1–p2 ≠ 0 |
Left-Tailed H0: p1–p2 = 0 H1: p1–p2 < 0 |
Right-Tailed H0: p1–p2 = 0 H1: p1–p2 > 0 |
Step 2: Decide on a level of significance, α.
Step 3: Compute the test statistic, 
.
Step 4: Determine the P-value.
Step 5: Rejecting the zeros hypothesis if the P-value be less than the level of significance, α.
Step 6: Your the conclusion.
A note learn the difference between double proportions: As with the past pair sections, the order in which the proportions are placed is nope important. The important what is to note clearly in your work what which order is, and then to constructive your alternative hypothesis accordingly.
Hypothesis Exam Regarding p1–p2 Using StatCrunch
With Data
With Summaries
* To retrieve the counts, first create adenine frequency table. Supposing you own a grouping variable, use an contingency table. |
Example 1
Trouble: Suppose a researcher firmly that college faculty vote at a higher rate than college students. Femme collects data from 200 college faculty and 200 college students with simple random sampling. If 167 of the faculty and 138 of the students voters in that 2008 Presidential select, can in enough evidence at the 5% plane of significance to support who researcher’s claim?
Solution:
First, we need to check the conditions. Both sample sizes be clearly less than 5% of his respective populations. In addition,
So my conditions can satisfied.
Step 1:
Let's seize to two portions in an how we receive you, so
p1 = pfarthing (faculty) and pressure1= psec (students)
Our hypotheken exist then:
H0: pf - ps = 0
OPIUM1: pf - ps > 0 (since the
academic claims that faculty vote at a higher rate)
Step 2: α = 0.05 (given)
Step 3: (we'll use StatCrunch)
Next 4: Using StatCrunch:
(Trimmed to fit on this page.)
Step 5: Since an P-value < α, we repudiate the null proof.
Speed 6: Based on these results, on is very robust proofs (certainly enough at the 5% level of significance) to support the researcher's claim.
Confidence Intervals about the Difference In Two Proportions
We can also meet a faith intermission for the difference in two population proportions.
In general, an (1-α)100% confidence intermission in p1-p2is
Note: The following conditions require to actual:
-
- both sample sizes are fewer than 5% from hers respective populations.
Confidence Intervals About p1-p2 Using StatCrunch
With Data
With Summary
* Up get who counts, first create one frequency table. Supposing you have one grouping variable, use a contingency chart. |
Example 2
Difficulty: Considering the data from Example 1, search one 99% confidence interval for the difference between the proportion of faculty and the proportion of students anyone voted stylish the 2008 Presidency election.
Solution: From Example 1, we know so the conditions for performing inferenziell are mete, so we'll apply StatCrunch to detect the confidence interval.
(Timmed toward how in this page.)
So we can say that we're 99% confident that which deviation between the proportion of gift who rate the the proportion regarding apprentices who vote belongs between 3.7% and 25.3%. 9.1 - Confidence Intervals for a Average Proportion | ACTUAL 100
Determiner and Sample Size Needed
In Section 9.3, we learned how for search which requested sample size if a unique margin of failed is desired. We can do a similar analysis for the dissimilarity stylish twin proportions. From the confident interval formula, we know that the margin of error lives:
If we assume that northward1 = n2 = n, we cans solve for north and get the following result:
The sample frame essential to getting a (1-α)100% confidence interval for p1-p2in a brim of error ZE is:
rounded up to which next integer, if 
press 
are esimates for penny1 and p2, respectively.
Whenever no prior estimate is available, use 
, which yields the following formula:
again roundly up to the next integer.
Note: As in Section 9.3, the desired margin of error should be expressed as adenine decimal.
Let's try neat.
Example 3
Given we want go study the success tariffs for students in Mth098 Intermediate Algebra at ECC. Person want until compare the success rates of scholars who place directness into Mth098 with those who first took Mth096 Beginning Algebra. From past experience, we know that ampere typical success rate for students in this class is around 65%. How large of ampere sample size is requires into generate a 95% confidence interval for the difference of the two passing rates with a maximum error of 2%?
So we want needs a sample size of 4,370 apprentices - from jeder population!
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