Exam 8: Hypothesis Testing

The following display from a TI-84 Plus calculator presents the results of a hypothesis test for a population mean $\mu$ . What is the value of $s$ ?

A test of $H _ { 0 } : \mu = 60$ versus $H _ { 1 } : \mu < 60$ is performed using a significance level of $\alpha = 0.01$ . The $P$ -value is $0.118$ . Is $H _ { 0 }$ rejected?

For the conjecture "The average age of students in this class is 22", the null hypothesis is:

Are the following statements $H _ { 0 } : = 12$ and $H _ { 1 } : \neq 12$ valid null and alternative hypotheses?

Science fiction novels average 290 pages in length. The average length of 10 randomly chosen novels written by I. M. Wordy was 365 pages in length with a standard deviation of 50 . At $\alpha = 0.05$ , are Wordy's novels significantly longer than the average science fiction novel?

Using the $z$ table, find the critical value (or values) for an $\alpha = 0.03$ two-tailed test.

A fleet of rental cars - all the same make, model, and year - has a mean fuel efficiency of $25.9$ miles per gallon (mpg). A random sample of 51 cars are selected and the air filter of each is replaced with a new one. Let $\mu$ be the population mean fuel efficiency score that would occur if every car's air filter were replaced. The air filter change is deemed effective if $\mu > 25.9$ mpg. A test is made of $H _ { 0 } : \mu = 25.9$ versus $H _ { 1 } : \mu > 25.9$ . Consider these possible conclusions: i). The air filter changes are effective. ii). The air filter changes are not effective. iii). The air filter changes might not be effective. Which of the three conclusions is best if $H _ { 0 }$ is not rejected?

A test of $H _ { 0 } : \mu = 56$ versus $H _ { 1 } : \mu \neq 56$ is performed using a significance level of $\alpha = 0.01$ . The value of the test statistic is $z = - 2.46$ . If the true value of $\mu$ is 56 , does the conclusion result in a Type I error, a Type II error, or correct decision?

In a simple random sample of size 100 , there were 25 individuals in the category of interest. Compute the sample proportion $\hat { p }$ .

$\text { A type I error occurs if one rejects the null hypothesis when it is }$ _______

The mean annual tuition and fees for a sample of 11 private colleges was $\$ 34,100$ with a standard deviation of $\$ 5400$ A dotplot shows that it is reasonable to assume that the population is approximately normal. You wish to test whether the mean tuition and fees for private colleges is different from $\$ 35,700$ . State a conclusion regarding $H _ { 0 }$ . Use the $\alpha = 0.10$ level of significance.

Determine whether the outcome is a Type I error, a Type II error, or a correct decision. A test is made of $H _ { 0 } : \mu = 40$ versus $H _ { 1 } : \mu \neq 40$ . The true value of $\mu$ is 40 and rejected.

Determine whether the outcome is a Type I error, a Type II error, or a correct decision. A test is made of $H _ { 0 } : \mu = 16$ versus $H _ { 1 } : \mu \neq 16$ . The true value of $\mu$ is 19 and not rejected.

A garden supplier claims that its new variety of giant tomato produces fruit with an mean weight of 46 ounces. A test is made of $H _ { 0 } : \mu = 46$ versus $H _ { 1 } : \mu \neq 46$ . The null hypothesis is not rejected. State the appropriate conclusion.

What are the critical value for a two-tailed $t$ test when $\alpha = 0.02$ and $n = 19$ ?

The formula for the test value in a z test for proportions is .

A test is made of $H _ { 0 } : \mu = 50$ versus $H _ { 1 } : \mu \neq 50$ . A sample of size $n = 71$ is drawn, and $\bar { x } = 56$ . The population standard deviation is $\sigma = 29$ . Compute the value of the test statistic $\mathrm { z }$ and determine if $H _ { 0 }$ is rejected at the $\alpha = 0.05$ level.

The following output from MINITAB presents the results of a hypothesis test for a population mean $\mu$ . Test of $\mathrm { mu } = 57$ vs. not $= 57$ Mean StDev SE Mean 95\% 45 60.26 11.575578 1.725585 (56.782306,63.737694) 1.889214 0.0654 Do you reject $H _ { 0 }$ at the $\alpha = 0.01$ level of significance? 4