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Given a random variable $X$ normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.

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Use the t-distribution to answer the questions below. You must give your answers correct to 3 decimal places.

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What are the values of t_low and t_high which between them contain the central 95% of values in the t- distribution, with:

(i) $\\var{df1}$ degrees of freedom

t_{low =[[0]]}

t_{high =[[1]]}

(ii) $\\var{df2}$ degrees of freedom

t_{low =[[2]]}

t_{high =[[3]]}

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What are the values of t_low and t_high which between them contain the central 99% of values in the t- distribution, with

(i) $\\var{df3}$ degrees of freedom

t_{low =[[0]]}

t_{high =[[1]]}

(ii) $\\var{df4}$ degrees of freedom

t_{low =[[2]]}

t_{high =[[3]]}

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i) What is the critical t-value associated with a sample of size $\\var{n1}$ and a significance level of 0.05 for a two-tailed distribution?

t_crit =[[0]]

ii) What is the critical t-value associated with a sample of size $\\var{n2}$ and a significance level of 0.01 for a one-tailed distribution?

t_crit =[[1]]

i) What is the value of $t_{{\\frac{\\alpha}{2},}{n-1}}$ where $n = \\var{n3}$ and ${\\alpha} = 0.05$?

$t_{{\\frac{\\alpha}{2},}{n-1}}$ = [[0]]

ii) What is the value of $t_{{\\frac{\\alpha}{2},}{n-1}}$ where $n = \\var{n4}$ and ${\\alpha} = 0.01$?

$t_{{\\frac{\\alpha}{2},}{n-1}}$ = [[1]]