Chapter 8 Wrap Up
Concept Check
Section Reviews
8.1 Inference for 2 Dependent Samples (Matched Pairs)
A hypothesis test for matched or paired samples (t-test) has these characteristics:
- Test the differences by subtracting one measurement from the other measurement
- Random Variable: = mean of the differences
- Distribution: Student’s-t distribution with n – 1 degrees of freedom
- If the number of differences is small (less than 30), the differences must follow a normal distribution.
- Two samples are drawn from the same set of objects.
- Samples are dependent.
Test Statistic (t-score): t = where: is the mean of the sample differences. μd is the mean of the population differences. sd is the sample standard deviation of the differences. n is the sample size.
8.2 Inference for 2 Independent Sample Means
A hypothesis test of two population means from independent samples where the population standard deviations are known (typically approximated with the sample standard deviations), will have these characteristics:
- Random variable: = the difference of the means
- Distribution: normal distribution
Normal Distribution:
.
Test Statistic (z-score):
And Null hypothesis Generally H0: µ1 – µ2 = 0 or µ1 = µ2
where:
σ1 and σ2 are the known population standard deviations. n1 and n2 are the sample sizes. and are the sample means. μ1 and μ2 are the population means.
However, most likely we do not know σ1 and σ2 so we usually use the t distribution with Test Statistic:
8.3 Inference for 2 Sample Proportions
Test of two population proportions from independent samples
- Random variable: p1-p2 , the difference between the two estimated proportions
- Distribution: normal distribution
We oftne use the “Pooled” Proportion:
pp =
Distribution for the differences:
With Test Statistic:
And Null hypothesis H0: pA = pB or H0: pA − pB = 0.
where:
p̂1 and p̂2 are the sample proportions, p1 and p2 are the population proportions,
Pp is the pooled proportion, and n1 and n2 are the sample sizes.
Key Terms
Try to define the terms below on your own. Scroll over any term to check your response!
8.1 Inference for 2 Dependent Samples (Matched Pairs)
- Placebo
- Inference
- Means
- Proportions
- Quantitative data
- Categorical data
- Independent
- Matched pairs
- Population mean difference
- Sampling distribution
- Point estimate
8.2 Inference for 2 Independent Sample Means
- Independent
- Quantitative data
- Parameter
- Difference in means
- Point estimate
- Standard error
- Inference
- Sampling distribution
- Test statistic
- Confidence interval
- Outliers
- Degrees of freedom
8.3 Inference for 2 Sample Proportions
- Categorical data
- Inference
- Independent
- Population proportion
- Sampling distribution
- Standard error
- Pooled proportion
- Confidence interval
- Point estimate
Extra Practice
8.1 Inference for 2 Dependent Samples (Matched Pairs)
1. Seven eighth graders at Kennedy Middle School measured how far they could push the shot-put with their dominant (writing) hand and th