Question Type 9: Initiating the one sample t-test with the hypothesis and critical region Exercises

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

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.

Explain the meaning of the significance level $\alpha=0.05$ and describe a Type I error in the context of testing the mean weight of bananas.

List the assumptions required for a valid one-sample t-test and comment on whether they are reasonable for the banana weight data.

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

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 $s$.

Using the t-statistic $t=-2.464$ and critical values $\pm2.201$, state your conclusion of the test at $\alpha=0.05$ (two-tailed).

Using the results $\bar x=80.75$, $s=13.01$, and $n=12$, calculate the t-statistic for testing $H_0:\mu=90$ against $H_1:\mu\neq90$.

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

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

At significance level $\alpha=0.01$, and with $t=-2.464$, $df=11$, what is your conclusion for a two-tailed test of $H_0:\mu=90$?

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