IB Mathematics Applications & Interpretation Question Type 11: Calculating Critical Value And/or P-value to Conclude the One Sample T-test Exercises

A chemical concentration test is performed with the following sample data: $n=14$, $\bar{x}=8.1$, and $s=0.9$. Test the null hypothesis $H_0:\mu =8$ against the alternative hypothesis $H_1:\mu >8$ at the $\alpha =0.05$ significance level using the $p$-value approach.

A study records $n=12$ observations with $\bar{x}=54$ and $s=9$. Test at the $\alpha =0.01$ significance level whether $\mu <60$. Use the $p$-value method.

A treatment yields $n=30$, $\bar{x}=15.5$, and $s=3.2$. Test at the 5% level whether the population mean equals 14 using the $p$-value approach.

A sample of size $n=10$ has mean $\bar{x}=52$ and standard deviation $s=4.5$. Test at the 5% significance level whether the population mean differs from $50$ using the critical value approach. Include hypotheses, critical value, test statistic, and conclusion.

A sample of $n=20$ yields $\bar{x}=7.4$ and $s=1.2$. Use the $p$-value approach at $\alpha =0.05$ to test $H_0:\mu =7$ versus $H_1:\mu \ne 7$. Compute the $p$-value and state your conclusion.

A manufacturer claims that their bulbs last on average 1200 hours. A quality control sample of $n=25$ bulbs gives $\bar{x}=1185$ and $s=40$. Test at $\alpha =0.10$ whether the true mean lifetime is less than 1200 hours using the critical value method.

A psychologist measures reaction times: $n=18$, $\bar{x}=250$ ms, $s=30$ ms. Test $H_0:\mu =260$ ms versus $H_1:\mu <260$ ms at $\alpha =0.05$ by finding the critical $t$-value and comparing.

A sample of $n=21$ yields $\bar{x}=100.8$ and $s=2.5$. Using the p-value approach at the $\alpha =0.01$ level of significance, test $H_0:\mu =102$ against $H_1:\mu <102$.

A sample of size $n=15$ yields $\bar{x}=103$ and $s=6$. At the 1% significance level, test whether the true mean exceeds 100 using the critical value approach. Provide all steps.

An educator tests if students' scores differ significantly from 75. Given $n=28$, $\bar{x}=78$, and $s=5$, test the hypothesis $H_0:\mu =75$ against $H_1:\mu \ne 75$ at the $1\%$ significance level ($\alpha =0.01$) using the critical value method.

In a lab, $n=8$ measurements give $\bar{x}=5.2$ and $s=0.8$. Test $H_0:\mu =5$ against $H_1:\mu \ne 5$ at $\alpha =0.05$. Find the critical values and conclude via comparison.