Comparing two population means - large independent samples

General Statistics

Statistical Inference
Comparing two population means

Comparing two population means - large independent samples

Statistics involving two populations proportions often have sample sizes that are large (), therefore the normal approximation distribution and associated statistics can be used to determine if or to test whether sample 1 proportion = sample 2 proportion.

That is, when the sample size is greater than or equal to 30 we can use the z-score statistics to compare the sample 1 proportion against sample 2 proportion using estimate of the sample proportion standard deviation,

The sample distribution of p (proportion) is approximately normal with a mean or expected value, E(P) = and standard error .

1. Know the statistics used to test two population proportions for large sample size.

For a two population proportions comparison the test statistics is related to the standard normal distribution:

Estimates of sample proportions, p₁ and p₂ are:
and
The pooled data proportion is:
and
Estimate of combined proportion standard deviation is:

The test statistics is:
, where
The difference of both proportion mean is:
(assume population proportions are the same)
Standard deviation:
Confidence Interval for is:

Decision rules:

Upper-Tailed Test ():

Accept H₀ if

Reject H₀ if

Lower-Tailed Test ():

Accept H₀ if

Reject H₀ if

Two-Tailed Test ():

Accept H₀ if

Reject H₀ if

Population or Sample Identification Sample size Number of Successes Population proportion

1 n₁

2 n₂

2. Know how to use appropriate statistics to test if two samples proportions are equal or if their difference = 0 (usually large sample size).

3 Types of tests in comparing two sample means:

When comparing the sample proportions, there are 3 questions to considered:

Question 1: : Is ? H_a (Two-tailed test)

Question 2: : Is ? H_a (Right-tailed test)

Question 3: : Is ? H_a (Left-tailed test)

Question 1: : Is ? H_a (Two-tailed test)

Is H_a
Are the sample proportions of both samples the same or not the same?
If there are the same then the difference between the two proportions will be close to 0.
Here we use a two-tailed test by computing the confidence interval for the test at the level of significance .
If the test statistics for z falls outside this interval we decide that the means differs, we chose the alternate hypothesis, H_a
Otherwise we have no reason to think that they differ, H₀

By Examples:

Problem 1. Test at the 95% confidence level if there is a difference between these two samples:

Sample
ID
Sample size Number of
Successes
Population
proportion

1 150 18 0.12

2 180 20 0.11

Given difference , n > 30 (large so can use normal approximation of z-score)

Step 1 - Hypothesis: The claim that , the null hypothesis.

The alternate hypothesis is that

H₀ :

H_a :

Step 2. Select level of significance: This is given as (5%)

So for two-tailed test:

Step 3. Test statistics and observed value.

&
The test statistics is:

Step 4. Determine the critical region (favors H_a)

For alpha = 0.05 at both ends of intervals: 0.025 and 0.975, z_a/2 = -1.96 and z_1-a/2 = 1.96 (from reference table)

The critical region is and

Step 5. Make decision.

No not reject the null hypothesis if or

The observed z = 0.25, and since 0.25 < 1.96 and is not in the critical region, we have no reason to reject H₀ in favor of H_a.

There the difference between both proportions are 0.

Question 2: : Is ? H_a (Right-tailed test)

IsH_a
Is the proportion of one sample greater than the proportion of another sample?
If one is smaller than the other, then their difference will be greater than or less than 0.
Here we use a one-tailed test by computing the confidence interval for the test at the level of significance .
If the test statistics for z falls outside this interval we decide that the means differs, we chose the alternate hypothesis, H_a
Otherwise we have no reason to think that they differ, H₀

By Examples:

Problem 1. Test at the 0.02 significance level whether sample proportion 1 is greater than sample proportion 2:

Sample
ID
Sample size Number of
Successes
Population
proportion

1 40 13 0.26

2 300 40 0.1333

Given difference , n > 30 (large so can use normal approximation of z-score)

Step 1 - Hypothesis: The claim that , the null hypothesis.

The alternate hypothesis is that

H₀ :

H_a :

Step 2. Select level of significance: This is given as (2%)

Step 3. Test statistics and observed value.

&
The test statistics is:

Step 4. Determine the critical region (favors H_a)

For alpha = 0.02, z_1-a/2 = 2.054 (from reference table)

The critical region is

Step 5. Make decision.

No not reject the null hypothesis if

The observed z = 2.313, and since 2.313 > 2.054 and is in the critical region, we reject H₀ in favor of H_a.

There the difference Sample 1 is greater than sample 2's proportion.

Question 3: : Is ? H_a (Left-tailed test)

IsH_a
Is the proportion of one sample less than the proportion of another sample?
If this is true then their difference will not be equal to 0.
Here we use a one-tailed test by computing the confidence interval for the test at the level of significance .
If the test statistics for z falls outside this interval we decide that the means differs, we chose the alternate hypothesis, H_a
Otherwise we have no reason to think that they differ, H₀

By Examples:

Problem 1. Test at the 0.10 significance level whether sample proportion 1 is greater than sample proportion 2:

Sample
ID
Sample size Number of
Successes
Population
proportion

1 300 40 0.1667

2 50 12 0.24