Supplementary Exercise 7.4 of IPS7e
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One-sample testing with a t-test of the null hypothesis
  H0: mu = 64 
against the alternative hypothesis
  Ha: mu <> 64 
based on a sample of 30 observations. Observed t-value is t_obs=1.12.

(a) The degrees of freedom are: df = n-1 = 30-1 = 29.

(b) Among the percentiles for a t(29)-distribution (a t-distribution
with 29 df) we find the following ones closest to the observed t_obs:
  t1=1.055 < t_obs=1.12 < t2=1.311

The right-tail (or upper-tail) probabilities for t1 and t2 are 0.15 and 
0.10, respectively, meaning that t1 and t2 are the 85% and 90% percentiles 
of the t(29)-distribution.

(c) The upper-tail probability for the observed t_obs=1.12 is now
bounded between 0.10 and 0.15, as shown below:
   0.15 > P(t(29)>t_obs) > 0.10,
and the P-value for the t-test against a two-sided alternative is twice
the tail probability. That is,
   0.30 = 2*0.15 > P > 2*0.10 = 0.20.
Of these bounds, the one of primary interest is that P>0.20.

(d) We saw above that P>0.20. Hence the test is not significant at the 10%
level and nor at the 5% level (in fact, not being significant at the 10% level,
means that it's not significant at any lower significance levels).

(e) Minitab gives the P-value as 
   P = 2*0.135952 = 0.272

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Minitab commands (to compute P-value and cutoffs) and output:

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 CDF -1.12;
  T 29.

Student's t distribution with 29 DF

    x  P( X <= x )
-1.12     0.135952

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 CDF 1.12 k1;
  T 29.
 let k2=2*(1-k1)
 Print K1 K2.

K1    0.864048
K2    0.271904

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 InvCDF .90;
  T 29.

Student’s t distribution with 29 DF

P( X <= x)        x
       0.9  1.31143

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 InvCDF .85;
   T 29.

Student’s t distribution with 29 DF

P( X <= x)        x
      0.85  1.05530