IB Mathematics Applications & Interpretation Question Type 14: Calculating the Critical Value And/or P-value to Conclude the Two Sample T-test Exercises

Two independent samples of exam scores were collected with the following results:

Sample 1: $n_1=25$, ${\bar{x}}_1=100$, $s_1=12$ Sample 2: $n_2=30$, ${\bar{x}}_2=95$, $s_2=10$

Assuming the populations are normally distributed with equal variances, perform a $t$-test at the $10\%$ significance level to determine if there is a difference between the mean scores of the two groups.

State the hypotheses, calculate the two-tailed $p$-value, and state your conclusion in context.

Two independent random samples are taken from populations with normal distributions and equal variances.

Sample 1: $n_1=15,{\bar{x}}_1=7.2,s_1=1.5$ Sample 2: $n_2=15,{\bar{x}}_2=6.8,s_2=1.2$

Test, at the 5% level of significance, the null hypothesis $H_0:{\mu }_1={\mu }_2$ against the alternative hypothesis $H_1:{\mu }_1>{\mu }_2$.

Calculate the pooled standard deviation estimate and the $t$-test statistic. Find the critical $t$-value and state your conclusion.

Perform a pooled $t$-test to determine if there is evidence at the $1\%$ significance level that ${\mu }_1>{\mu }_2$. Calculate the $p$-value and state your conclusion.

The lifetimes of two types of industrial components were measured to compare their mean performance.

Sample 1 consisted of $n_1=18$ components with a mean lifetime ${\bar{x}}_1=120$ hours and a standard deviation $s_1=15$ hours.

Sample 2 consisted of $n_2=16$ components with a mean lifetime ${\bar{x}}_2=110$ hours and a standard deviation $s_2=20$ hours.

Assuming the lifetimes of both populations follow a normal distribution with equal variances, test the hypothesis $H_0:{\mu }_1={\mu }_2$ against $H_1:{\mu }_1>{\mu }_2$ at the $5\%$ level of significance. Calculate the one-tailed $p$-value and state the conclusion of the test.

Perform a two-tailed $t$-test at the $5\%$ significance level to determine whether there is a significant difference between the population means ${\mu }_1$ and ${\mu }_2$.

Two samples of weights provide the following data: Sample 1: $n_1=8$, ${\bar{x}}_1=85$, $s_1=6$ Sample 2: $n_2=10$, ${\bar{x}}_2=80$, $s_2=5$

At the $1\%$ significance level, test for a difference in means (two-tailed). Find the critical $t$-value and conclude your findings.

A researcher compares the average reaction times of two groups. Sample 1: $n_1=12$, ${\bar{x}}_1=24$ ms, $s_1=4$ ms; Sample 2: $n_2=15$, ${\bar{x}}_2=20$ ms, $s_2=5$ ms. At the 5% significance level, test for a difference in means by finding the critical $t$-value and stating your conclusion.

Two independent samples are taken from normally distributed populations with equal variances. The sample statistics are as follows:

Sample 1: $n_1=20$, ${\bar{x}}_1=200$, $s_1=30$ Sample 2: $n_2=22$, ${\bar{x}}_2=185$, $s_2=25$

At the 1% significance level, test the hypothesis $H_0:{\mu }_1={\mu }_2$ against the alternative hypothesis $H_1:{\mu }_1>{\mu }_2$.

Find the critical $t$-value and state your decision.

Independent samples are taken from two normal populations with equal variances. The sample statistics are as follows:

Sample 1: $n_1=30,{\bar{x}}_1=54,s_1=8$ Sample 2: $n_2=28,{\bar{x}}_2=50,s_2=7$

A two-tailed $t$-test is conducted at the 5% level of significance to determine if the population means differ significantly. Calculate the pooled standard deviation and the test statistic. Determine the critical $t$-value and state the conclusion of the test, giving a reason for your answer.

Two independent samples yield the following data: Sample 1: $n_1=20,{\bar{x}}_1=15,s_1=3$ Sample 2: $n_2=18,{\bar{x}}_2=12,s_2=4$

Test $H_0:{\mu }_1={\mu }_2$ against $H_1:{\mu }_1>{\mu }_2$ at the $5\%$ significance level. Calculate the $t$-statistic, the one-tailed $p$-value, and state your conclusion.

Two independent samples are taken from normally distributed populations with equal variances. The sample statistics are as follows:

Sample 1: $n_1=16$, ${\bar{x}}_1=250$, $s_1=30$ Sample 2: $n_2=14$, ${\bar{x}}_2=230$, $s_2=25$

Test the hypothesis $H_0:{\mu }_1={\mu }_2$ against $H_1:{\mu }_1\ne {\mu }_2$ at the $5\%$ significance level. Calculate the pooled standard deviation, the $t$-statistic, and the $p$-value to justify your conclusion.

At the $5\%$ significance level, find the critical $t$-value and state the conclusion for a two-tailed test of the null hypothesis $H_0:{\mu }_1={\mu }_2$ against the alternative hypothesis $H_1:{\mu }_1\ne {\mu }_2$.