IB Mathematics Applications & Interpretation Question Type 9: Initiating the One Sample T-test with the Hypothesis and Critical Region Exercises

Using the results $\bar{x}=80.75$, $s=13.01$, and $n=12$, calculate the $t$-statistic for testing $H_0:\mu =90$ against $H_1:\mu \ne 90$.

List the assumptions required for a valid one-sample $t$-test and comment on whether they are reasonable for this study.

For a two-tailed one-sample $t$-test with $n=12$ at a significance level of $\alpha =0.05$, find the critical $t$-value(s).

Construct a 95% confidence interval for the true mean weight of bananas given $\bar{x}=80.75$, $s=13.01$, and $n=12$.

A new sample of $20$ bananas has $\bar{x}=88$ and $s=12$. Perform a two-tailed one-sample $t$-test at the $5\%$ significance level ($\alpha =0.05$) to test $H_0:\mu =90$. Compute the $t$-statistic and state your conclusion.

A two-tailed hypothesis test is performed for $H_0:\mu =90$ at a significance level of $\alpha =0.01$.

The test statistic is found to be $t=-2.464$ with $11$ degrees of freedom.

Determine the conclusion of the test, justifying your answer.

State the null and alternative hypotheses for a one-sample $t$-test to determine whether the average weight of bananas differs from 90 grams.

Given the banana weights (in grams): 80, 100, 60, 70, 75, 86, 75, 91, 84, 84, 63, 101, calculate the sample mean $\bar{x}$ and sample standard deviation $s$.

Explain the meaning of the significance level $\alpha =0.05$ and describe a Type I error in the context of this test.

State the null and alternative hypotheses for a one-sample $t$-test to determine whether the average weight of bananas is less than $90$ grams.

For the t-statistic $t=-2.464$ with $df=11$, calculate the two-tailed p-value.

Using the $t$-statistic $t=-2.464$ and the critical values $\pm 2.201$, state the conclusion of the test.