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1. What is the critical region for a 5% two-sided test if, under the null hypothesis, the test statistic is distributed as \text{Student}(0,\sigma,4)? That is, what values of the test statistic disprove the null hypothesis with p < 0.05? (OK to use Python, MATLAB, or Mathematica.)

from scipy.stats import t
print t.ppf(.025, 4)
print t.ppf(.975, 4)
value 1: -2.7764451052
value 2: 2.7764451052

2. For an exponentially distributed test statistic with mean \mu (under the null hypothesis), when is the the null hypothesis disproved with p < 0.01 for a one-sided test? for a two-sided test?

Failed to parse (unknown error): p = \lambda{e^{-\lambda{x}}}

Failed to parse (unknown error): \mu = \lambda^{-1}


Failed to parse (unknown error): \int_0^{x}\lambda{e^{-\lambda{x}}}\,dx = .99

Failed to parse (unknown error): 1-e^{-\lambda{x}}=.99

Failed to parse (unknown error): x = \frac{-ln(.01)}{\lambda} = -\mu*{ln(.01)}

In two sided tests with p = .01, we must find two x's

Failed to parse (unknown error): \int_0^{x}\lambda{e^{-\lambda{x}}}\,dx = .995

Failed to parse (unknown error): 1-e^{-\lambda{x}}=.995

Failed to parse (unknown error): x = \frac{-ln(.005)}{\lambda} = -\mu*{ln(.005)}

Failed to parse (unknown error): \int_0^{x}\lambda{e^{-\lambda{x}}}\,dx = .005

Failed to parse (unknown error): 1-e^{-\lambda{x}}=.005

Failed to parse (unknown error): x = \frac{-ln(.995)}{\lambda} = -\mu*{ln(.995)}