Question Type 1: Formulating test hypothesis for different types of tests Exercises

Formulate the null and alternative hypotheses in symbols for a test to determine whether the mean waiting time exceeds 30 minutes.

Formulate the null and alternative hypotheses in symbols for testing whether the average concentration differs from 5 mg/L.

Formulate the null and alternative hypotheses in words and symbols for a test to determine whether the average score is less than 80.

A machine is set to fill jars with jam to a target weight of 500 g. Formulate the null and alternative hypotheses in symbols to test whether the machine is underfilling.

For $X\sim N(5,9)$, find the critical value $c$ such that $P(X>c)=0.07$.

For a significance level of 7%, determine the rejection region for $\bar X$ when sampling $n=36$ from $N(\mu=5,\sigma^2=9)$ for a one-tailed test $H_1:\mu>5$.

Given $X\sim N(5,9)$ and an observed value $x=8$, calculate the p-value for the two-tailed test of $H_0:\mu=5$.

Determine the critical value $\bar x_c$ for which $P(\bar X<\bar x_c)=0.07$ given $\bar X\sim N(5,9/36)$.

Given $X\sim N(5,9)$ and observed $x=2$, compute the p-value for the one-tailed test $H_1:\mu<5$.

For $X\sim N(5,9)$, find the critical values $c_1$ and $c_2$ such that $P(X<c_1\text{ or }X>c_2)=0.07$ for a two-tailed test.

A sample of size $n=25$ from a normal population with known $\sigma=3$ yields a sample mean $\bar x=7$. Under $H_0:\mu=5$, calculate the p-value for a two-tailed test.

Given the critical value $\bar x_c=4.26$ and an observed $\bar x=4.20$ from a sample of size 36, calculate the p-value for $H_1:\mu<5$.