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

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

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

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 15 students is taken from a population where student weights are normally distributed. The weights $x_i$, in kg, of these students are summarized by $\sum_{i=1}^{15} x_i = 1095$ and $\sum_{i=1}^{15} x_i^2 = 80100$.

Test, at the $5\%$ level of significance, whether the population mean weight is less than 75 kg.

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.

The speeds (km/h) of 16 cars passing a checkpoint are recorded as follows: $58, 62, 61, 59, 63, 60, 64, 58, 59, 62, 61, 60, 63, 59, 62, 60$

Test at the $\alpha = 0.05$ significance level whether the mean speed of cars is equal to 60 km/h.

The heights, in cm, of 12 randomly selected students are as follows:

$172, 168, 169, 171, 170, 173, 174, 167, 175, 169, 171, 170$

Test, at the $5\%$ level of significance, whether the population mean height differs from $170\text{ cm}$.

Given $n = 30$, $\bar{x} = 5.2$ and $s^2 = 0.49$, test at the 1% significance level ($\alpha = 0.01$) whether the population mean differs from 5.

Test, at the $\alpha = 0.01$ significance level, whether the population mean temperature differs from $20^{\circ}\text{C}$.

A sample of 10 daily temperatures (in $^{\circ}\text{C}$) is recorded as follows:

$19.3, 21.5, 22.5, 18.9, 23.4, 18.2, 19.6, 24, 23, 24.4$

Test at the $\alpha=0.05$ significance level whether the mean daily temperature is larger than $23.5^{\circ}\text{C}$.

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.

A data set of size $n=5$ is given as follows: $2.1, 2.5, 2.3, 2.2, 2.4$.

Test, at the $5\%$ level of significance, whether the population mean $\mu$ exceeds $2$. Assume the population is normally distributed.