An independent random sample is drawn from a normally distributed population of unknown variance. Determine the p-values for the following hypotheses and t-stats, and specify whether the hypothesis to the level $\alpha = 5\%$ is rejected.
a. $H_1: \mu>\mu_0, \ n=211, \ t=1.91$
b. $H_1: \mu<\mu_0, \ n=171, \ t=-3.45$
c. $H_1: \mu\neq \mu_0, \ n=1704, \ t=0.83$
d. $H_1: \mu>\mu_0, \ n=2104, \ t=2.13$
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The p-value is defined as the probability, under the null hypothesis, of obtaining a result equal to or more extreme than what was actually observed.
At the alternative hypothesis we have that what we've observed, or not?
How can we calculate that probability?
Do we maybe calculate for example at a. the probability $P(t>\mu_0)$ ?
All of these sample sizes are large enough that the t statistic is very nearly normal. You don't give any evidence of engagement here. Are you supposed to use a normal approximation of software? A printed t table will be essentially useless. I will give hints to get you started.
For (b): in R statistical software
pt(-3.45, 170)returns p-value $0.0003535081 < 0.05,$ so reject.In (a), you'd reject at the 5% level, but not the 2% level. (Look at right-tail probability.)
For the two-tailed test in (c), you'll have to add areas from both tails.