10 Inferences for Two Population Means

   Population 1
      Sample 1
        		    Population 2
      Sample 2
        

Theorem: Sampling Distribution of the difference between two means (Large and independent samples):
Suppose independent random samples of size n1 and n2 are taken from two populations with means m1 and m2 and standard deviations s1 and s2. Then is approximately normally distributed and has mean = and standard deviation = The standardized random variable, z = , has approximately the standard normal distribution.

Note: If , then z = ( this can serve as the test statistic for hypothesis testing)

Type of Hypothesis Test: Two-Sample z-Test for Two Population Means
Assumptions:
1. independent samples
2. large samples (n> 30)
    i. State Ho and Ha
    ii. Decide on significance level, a
    iii. Find the critical values (±za/2, -za, za)
    iv. Compute the test statistic (z = )
    v. Reject or do not reject Ho
    vi. State your conclusion

Confidence Interval
To determine a confidence interval for the difference between two population means, use the endpoints:


Theorem: Sampling Distribution of the difference between two means (Normal Populations, independent samples):
Suppose independent random samples of size n1 and n2 are taken from two normal populations with means m1 and m2 and standard deviations s1 and s2. Then is also normally distributed and has mean = and standard deviation . The standardized random variable, z = , has the standard normal distribution.


Pooled sample standard deviation:

Note: has the t-distribution with df = n1 + n2 - 2

Type of Hypothesis Test: Pooled t-test for Two Population Means
Assumptions:
1. independent samples
2. large samples (n ≥ 30)
3. "equal" standard deviations
    i. State Ho and Ha
    ii. Decide on significance level, a
    iii. Find the critical values (±ta/2, -ta, ta)
    vi. Compute the test statistic (t = )
    v. Reject or do not reject Ho
    vi. State your conclusion
(Remember the P-test)

Confidence Interval
To determine a confidence interval (pooled t-interval procedure) for the difference between two population means ( ), use the endpoints:
   with df = n1 + n2 - 2.


Theorem: Distribution of non-pooled t-statistic (Normal Populations, independent samples):
Suppose independent random samples of size n1 and n2 are taken from two normal populations with means m1 and m2 and standard deviations s1 and s2. Then the random variable, t = , has approximately the t-distribution, with df = D (rounded down)

Type of Hypothesis Test: Non-Pooled t-test for Two Population Means
Assumptions:
1. independent samples
2. Normal Populations
    i. State Ho and Ha
    ii. Decide on significance level, a
    iii. Find the critical values (±ta/2 , -ta, ta ) with D degrees of freedom
    iv. Compute the test statistic (t = )
    v. Reject or do not reject Ho
    vi. State your conclusion
(Remember the P-test)

Confidence Interval
To determine a confidence interval (non-pooled t-interval procedure) for the difference between two population means (), use the endpoints:
     with df    =          (rounded down)

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