IB Mathematics Applications & Interpretation 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 $H_1:\mu \ne 52$ 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.