Question

question 3
1 pts
for the hypothesis test \\(\

$$\begin{aligned}h_0&:\\mu = 100\\\\h_a&:\\mu \ eq 100\\end{aligned}$$

\\), where \\(\sigma = 28\\), \\(\alpha = 0.10\\), and a sample of 64 had \\(\bar{x} = 107\\), find the \\(p\\)-value.
\\(\

$$\begin{aligned}&.9772\\\\&.0228\\\\&.0456\\\\&1.28\\\\&2\\\\&1.645\\end{aligned}$$

\\)
question 4
1 pts
for the hypothesis test \\(\

$$\begin{aligned}h_0&:\\mu \\leq 60\\\\h_a&:\\mu > 60\\end{aligned}$$

\\), where \\(\sigma = 6\\) and a sample of 49 had \\(\bar{x} = 60.9\\), and \\(\alpha = 0.01\\), find the critical value(s).
\\(\

$$\begin{aligned}&2.575\\\\&2.33\\\\&1.05\\\\&\\pm 2.575\\\\&\\pm 2.33\\\\&\\pm 1.05\\end{aligned}$$

\\)
question 5
1 pts

Explanation:

Question 3

Step1: Calculate the test statistic z

The formula for the z - statistic in a hypothesis test for the population mean (when $\sigma$ is known) is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$.
Given $\mu = 100$, $\bar{x}=107$, $\sigma = 28$, $n = 64$.
First, calculate the standard error $\frac{\sigma}{\sqrt{n}}=\frac{28}{\sqrt{64}}=\frac{28}{8}=3.5$.
Then, $z=\frac{107 - 100}{3.5}=\frac{7}{3.5}=2$.

Step2: Find the p - value for a two - tailed test

For a two - tailed test with $H_{a}:\mu
eq100$, the p - value is $2\times(1 - P(Z < |z|))$.
We know that $z = 2$, and from the standard normal table, $P(Z < 2)=0.9772$.
So, $1 - P(Z < 2)=1 - 0.9772 = 0.0228$.
Then the p - value is $2\times0.0228 = 0.0456$.

Step1: Determine the type of test and the critical value approach

We have a right - tailed test ($H_{a}:\mu>60$) with $\alpha = 0.01$ and the population standard deviation $\sigma$ is known. So we use the z - distribution.

Step2: Find the critical value

For a right - tailed test with $\alpha=0.01$, we need to find the z - value such that $P(Z > z_{\alpha})=\alpha = 0.01$, or $P(Z < z_{\alpha})=1-\alpha=0.99$.
From the standard normal table, the z - value corresponding to a cumulative probability of 0.99 is $z = 2.33$ (more precisely, $z = 2.326\approx2.33$).

Answer:

0.0456

Question 4