Question Type 11: Calculating critical value and/or p-value to conclude the one sample t-test Exercises

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.

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.

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 chemical concentration test: $n=14$, $\bar x=8.1$, $s=0.9$. Test $H_0:\mu=8$ vs $H_1:\mu>8$ at $\alpha=0.05$. Use the p-value approach.

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.

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

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 $H_1:\mu\neq7$. Compute the p-value and conclude.

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

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

An educator tests if students score differently from 75: $n=28$, $\bar x=78$, $s=5$. At $\alpha=0.01$, test $H_0:\mu=75$ vs $H_1:\mu\neq75$ using the critical value method.