Question Type 1: Performing a hypothesis test for the mean given the variance for single sample Exercises

A sample of size $n=12$ from a normal population has known variance $\sigma^2=4$ and yields observations: 15.5, 16.2, 15.8, 16.7, 16.0, 15.9, 16.3, 16.1, 15.7, 16.4, 15.6, 16.5. Test $H_0:\mu=16$ against $H_1:\mu\neq16$ at the 5% level.

For the same sample and known variance $\sigma^2=5$, test at the 5% level whether the mean temperature is greater than 25°C.

With the same data and $\sigma^2=4$, test $H_0:\mu=16$ vs $H_1:\mu<16$ at the 5% significance level.

The daily temperature in a city during summer follows a normal distribution with known variance $\sigma^2 = 5$. A sample of size 10 yields temperatures: 23, 21, 22.5, 24, 26, 22, 24, 24.5, 23.5, 27. Test whether the mean temperature differs from 25°C at the 5% significance level.

Using the same sample, test $H_0:\mu=16$ vs $H_1:\mu<16$ at the 5% significance level.

Temperatures: 19.3, 21.5, 22.5, 18.9, 23.4, 18.2, 19.6, 24, 23, 24.4. Assume daily temperature is normal with unknown variance. Test whether the mean exceeds 23.5°C at $\alpha=0.05$.

A sample of size 12: 15.2, 16.8, 17.1, 14.9, 16.4, 17.5, 16.0, 15.6, 17.2, 16.9, 15.8, 16.3. Assume normal with unknown variance. Test $H_0:\mu=16$ vs $H_1:\mu\neq16$ at $\alpha=0.05$.

Using the same data and known variance $\sigma^2=5$, test whether the mean temperature differs from 25°C at the 1% significance level.

With the same data ($n=12$), test $H_0:\mu=16$ vs $H_1:\mu\neq16$ at the 1% level.

Using the same sample ($n=12$, $\sigma^2=4$), test $H_0:\mu=16$ vs $H_1:\mu\neq16$ at the 1% level.

Using the same data with unknown variance, test $H_0:\mu=23.5$ vs $H_1:\mu\neq23.5$ at the 5% level.

A sample of size 15 yields temperatures: 18.7, 19.5, 20.2, 21.1, 19.8, 20.5, 20.0, 19.3, 21.4, 20.9, 19.6, 18.9, 20.3, 21.0, 19.7. Assume normal with unknown variance. Test $H_0:\mu=20$ at $\alpha=0.05$ (two-tailed).