t-interval for paired data

Inference for Numerical Data in R

Mine Cetinkaya-Rundel

Associate Professor of the Practice, Duke University

High School and Beyond

200 observations were randomly sampled from the High School and Beyond survey.The same students took a reading and writing test. At a first glance, how are the distributions of reading and writing scores similar? How are they different?

chp2-vid2-hsb2

Inference for Numerical Data in R

Independent scores?

Can reading and writing scores for a given student student assumed to be independent of each other?

Probably not!

Inference for Numerical Data in R

Analyzing paired data

  • When two sets of observations have this special correspondence (not independent), they are said to be paired.

  • To analyze paired data, it is often useful to look at the difference in outcomes of each pair of observations: diff = read ? write.

student read write diff
1 57 52 5
2 68 59 9
3 44 33 11
... ... ... ...
200 63 65 -2
Inference for Numerical Data in R

Estimating the mean difference in paired data

Construct a 95% confidence interval for the mean difference between the average reading and writing scores.

Inference for Numerical Data in R

Estimating the mean difference in paired data

Construct a 95% confidence interval for the mean difference between the average reading and writing scores.

t.test(hsb2$diff, conf.level = 0.95)
Inference for Numerical Data in R

Estimating the mean difference in paired data

Construct a 95% confidence interval for the mean difference between the average reading and writing scores.

t.test(hsb2$diff, conf.level = 0.95)
    One Sample t-test
data:  hsb2$diff
t = -0.86731, df = 199, p-value = 0.3868
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -1.7841424  0.6941424
sample estimates:
mean of x 
   -0.545 
Inference for Numerical Data in R

Interpreting the CI for mean difference in paired data

95% CI for the mean difference in reading and writing scores (read - write) is (-1.78, 0.69)

vs.

We are 95% confident that the average reading score is 1.78 points lower to 0.69 points higher than the average writing score.

Inference for Numerical Data in R

Let's practice!

Inference for Numerical Data in R

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