IB Mathematics Applications & Interpretation Question Type 1: Formulating Test Hypothesis for Different Types of Tests Exercises

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.

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

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

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 an observed value $x=2$, find the $p$-value for the one-tailed test $H_1:\mu <5$.

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 $H_1:\mu >5$.

For an observed sample mean of $\bar{x}=4.20$, calculate the $p$-value for the test.

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

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

For $X\sim N(5,9)$, find the critical 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.