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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 $P(t>\mu_0)$ ?

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  • $\begingroup$ You've been around this site long enough not to be surprised by the votes to 'close'. None of them mine. $\endgroup$
    – BruceET
    Apr 18, 2018 at 16:10

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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.

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  • $\begingroup$ If the standard deviation $ \sigma $ of the population is unknown, we use the t-distribution. The t-distribution approximates the normal distribution for large $ n $, for example, for $ n> 30$. So, that means that in every case we consider the normal distribution, right? Let's consider (a). Is the p-value defined as $P(T<t \mid \mu=\mu_0)$ ? $\endgroup$
    – Mary Star
    Apr 18, 2018 at 19:31
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    $\begingroup$ Some people wrongly say to use z instead of t if n > 30. That aprx may be OK for 5% level, but not for 1% or 10%. // But your sample sizes are all so big that normal aprx would be OK for these drill exercises. // If using real data in statistical software pkg, always use t if $\sigma$ unknown because z procedure would ask for unknown $\sigma.$ $\endgroup$
    – BruceET
    Apr 18, 2018 at 19:46
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    $\begingroup$ Assuming $H_0$ true, p-value is probability of a value more extreme (in direction or directions of $H_a$) than the observed value of the test statistic. Thus in (b) it's the probability below -3.45 in the left tail of $\mathsf{T}(df=170).$ $\endgroup$
    – BruceET
    Apr 18, 2018 at 21:16
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    $\begingroup$ Not absolutely sure of your notation; (c) makes no sense. For that add two tail probabilities. Maybe you'd denote it as $P(T < -t)+P(T > t).$ Anyhow, hugely non-significant. $\endgroup$
    – BruceET
    Apr 19, 2018 at 6:46
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    $\begingroup$ Grading exams for a couple of days. Hope someone else can help. $\endgroup$
    – BruceET
    Apr 19, 2018 at 18:20

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