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.