# Problem 1 Imagine you perform preliminary &quot;pilot' study t...

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(a) If $z = \frac{\bar x - 300}{\sigma/\sqrt{n}}$ and $\alpha = 0.05$, for what values of $z$ would you reject the null hypothesis at the $\alpha$ level? (Be sure to consider the alternative in finding your answer).
{r}
qnorm(0.05, lower.tail = FALSE)

(b) If the true population standard deviation is $\sigma = 142$ and $\mu = 300$ and $n = 100$, for what values of $\bar x$ will you reject the null hypothesis $H_0$?
If the $\bar x$ is greater than the following value:
{r}
sigma <- 142; mu <- 300; n <- 100
xbar <- qnorm(0.05, lower.tail = FALSE)*sigma/sqrt(n)+mu
print(xbar)

(c) If the true population standard deviation is $\sigma = 142$ and $\mu = 315$ and $n = 100$, what is the probability that $\bar x$ will fall into the region defined in (b)?
{r}
sigma <- 142; mu <- 315; n <- 100
z <- (xbar - mu)/(sigma/sqrt(n))
print(pnorm(z,lower.tail=FALSE))

(d) If you want 80% power to reject the null hypothesis, when the truth is that $\mu = 315$, do you achieve that goal? What can you do to increase the amount of power you have?
The power of a test is the probability of rejecting $H_0$ when $H_1$ is true. The solution to the
previous questions is actually the power of a test given $\alpha=.05$, 0.2780931 < 0.80. Increasing the sample size n help increase the power...

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