Question Type 2: Performing a hypothesis test for the mean with unknown variance of a single sample Exercises

Given completion times (hours) for 8 projects: $10.2,11.5,12.3,9.8,10.7,11.2,12.0,10.4$, test at $\alpha=0.10$ whether the mean time is below 12 hours.

A sample of 20 measurements yields $\bar x=34.8$ and sample variance $s^2=9.61$. Test at $\alpha=0.01$ if the true mean exceeds 33.

Data: $2.1,2.5,2.3,2.2,2.4$ ($n=5$). Test at $\alpha=0.05$ whether the mean exceeds 2.

A sample of 15 students has total weight $\sum x_i=1095$ kg and $\sum x_i^2=80100\,\text{kg}^2$. Test at $\alpha=0.05$ whether the mean weight is less than $75$ kg.

Heights (cm) of 12 students: $172,168,169,171,170,173,174,167,175,169,171,170$. Test at $\alpha=0.05$ whether their mean height differs from 170 cm.

Test at $\alpha=0.05$ whether the mean daily temperature is larger than $23.5^{\circ}\text{C}$ given the sample of 10 days: $19.3,21.5,22.5,18.9,23.4,18.2,19.6,24,23,24.4$.

Using the same 10-day temperature sample ($19.3,21.5,22.5,18.9,23.4,18.2,19.6,24,23,24.4$), test at $\alpha=0.01$ whether the mean differs from $20^{\circ}\text{C}$.

A sample of 12 observations yields $\bar x=1000$ and $s=30$. Test at $\alpha=0.01$ if the true mean exceeds 980.

Speeds (km/h) of 16 cars: $58,62,61,59,63,60,64,58,59,62,61,60,63,59,62,60$. Test at $\alpha=0.05$ whether the mean speed equals 60 km/h.

A sample of 10 items has $\bar x=50$ and $s=4.2$. Test $H_0:\mu=52$ versus $H_1:\mu\neq52$ at $\alpha=0.05$ and compute the $p$-value.

For $n=30$, $\bar x=5.2$ and $s^2=0.49$, test at $\alpha=0.01$ whether the mean differs from 5.

A sample of 25 readings has $\bar x=102$ and $s=15$. Test at $\alpha=0.05$ if the true mean is greater than 100, and find a 95% confidence interval.