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M241
Student t
Population
⇑
Parameter
µ
1. An approximate 95% confidence interval for µ:
November 4
=⇒
x1 , . . . , x n
⇐=
x
Sample
⇓
Statistic
s
x̄ ± 1.96 √ .
n
(a) Approxmation 1 – x̄ has normal distribution. This is exacty true only if the population is normal and it is
only approximately true if n is large or the population is close to normal
(b) Approximation 2 – σ = s
2. Student (William Sealey Gosset) noticed that the confidence intervals generated by the procedure above seemed
too small, especially for small samples, even if the population is exactly normal.
3. Suppose the population is normal:
x̄ − µ
√ has a standard normal distribution
σ/ n
x̄ − µ
√ has a t-distribution with parameter n − 1 (called degrees of freedom)
(b)
s/ n
(a)
4. If the population is normal, a 95% confidence interval for µ is
s
x̄ ± t∗ √ where t∗ is the critical value coming
n
from the t-distribution.
5. To predict the value ofra new x, use a prediction interval. A 95% prediction interval for the value of a new
1
measurement x is x̄ ± s 1 + .
n
> qt(.975,29)
[1] 2.045230
> t.test(dimes$Mass)
> t.test(dimes$Mass,conf.level=.99)
One Sample t-test
One Sample t-test
data: dimes$Mass
t = 560.4393, df = 29, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
2.249992 2.266474
sample estimates:
mean of x
2.258233
Homework
1. Read Devore and Farnum pages 313–318.
2. Do problems 7.4.38,39.
data: dimes$Mass
t = 560.4393, df = 29, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
99 percent confidence interval:
2.247127 2.269340
sample estimates:
mean of x
2.258233