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 $H_1: \mu \neq 7$. 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 $H_1: \mu \neq 75$ 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 $H_1:\mu\neq5$ at $\alpha=0.05$. Find the critical values and conclude via comparison.