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Chapter 16
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Using two different values of n (each greater than 3) and the normal distribution for
the errors, demonstrate
a) that the sample slope is normally distributed
b) that the test statistic, (b1 − null)/estSE, is t distributed
c) that, as n increases, the test statistic, (b1 − null)estSE, gets closer and closer
to the normal curve.
Put your results into a Word or Excel document copying the pictures of the Monte
Carlo experiments with the Take a Picture of the Results button and adding some
explanatory text.
a) That the sample slope is normally distributed
Summary Statistics
Average
4.9973
SD
0.4479
Max
6.9845
Min
3.1621
Statistic
Error Box
SD Errors
n
Normal Tails
Slope
Normal
1
5
10.0%
Summary Statistics
Average
4.9986
SD
0.2592
Max
6.0948
Min
3.9896
Empirical Histogram for 30000 Repetitions
3
4
5
6
Statistic
Error Box
SD Errors
n
Normal Tails
Slope
Normal
1
15
10.1%
Empirical Histogram for 30000 Repetitions
7
3
sample slope estimates
4
5
6
7
sample slope estimates
In each case, the normal curve has been superimposed over the sampling distribution.
This is strong evidence that the sample slope is normally distributed when the errors are
normal regardless of the sample size.
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Answers
Chapter 16
b) that the test statistic, (b1 − null)/estSE, is t distributed
c) that, as n increases, the test statistic, (b1 − null)estSE, gets closer and closer
to the normal curve.
Summary Statistics
Average
0.0038
SD
1.6951
Max
24.2476
Min
-18.2933
Statistic
Error Box
SD Errors
n
Normal Tails
Test Stat
Normal
5
5
19.9%
Summary Statistics
Average
-0.0021
SD
1.0919
Max
6.4611
Min
-6.1211
Empirical Histogram for 10000 Repetitions
-10
-5
0
5
Statistic
Error Box
SD Errors
n
Normal Tails
Test Stat
Normal
5
15
12.4%
Empirical Histogram for 10000 Repetitions
10
-10
test stat values
-5
0
5
10
test stat values
The two pictures above, at two values of n, demonstrate both that the t-distribution works
better for smaller sample probability histograms and that the t-distribution converges to
the normal as the sample size increases. On the left, we see that the normal distribution
over estimates the probability near 0, whereas it underestimates the probability near the
tails. The t-distribution follows the Monte Carlo histogram almost exactly. On the right,
the Normal, the t, and the Monte Carlo histogram have all converged.
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