Exam 8: Hypothesis Testing

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion. -In tests of a computer component, it is found that the mean time between failures is 520 hours. A modification is made which is supposed to increase the time between failures. Tests on a random sample of 10 modified components resulted in the following times (in hours)between failures. 518 548 561 523 536 499 538 557 528 563 At the 0.05 significance level, test the claim that for the modified components, the mean time between failures is greater than 520 hours.

Solve the problem. -Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense. A math teacher tries a new method for teaching her introductory statistics class. Last year the mean score on the final test was 73. This year the mean on the same final was 76.

Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither. -Claim: $\mu = 959$ . Sample data: $n = 25 , \bar{x} = 951 , s = 25$ . The sample data appear to come from a normally distributed population with $\sigma = 28$ .

Determine whether the given conditions justify testing a claim about a population mean µ -The sample size is $n = 17$ , $\sigma$ is not known, and the original population is normally distributed.

$\text { Find the value of the test statistic } z \text { using } z = \frac { \hat { p } - p } { \sqrt { \frac { p q } { n } } } \text {. }$ -A claim is made that the proportion of children who play sports is less than 0.5, and the sample statistics include n = 1158 subjects with 30% saying that they play a sport.

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. -An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Littleton. A random sample of 234 fathers from Littleton yielded 96 who did not help with child care. Test the researcher's claim at the 0.05 significance level.

Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense. Of a group of 1000 people suffering from arthritis, 500 receive acupuncture treatment and 500 receive a placebo. Among those in the placebo group, 24% noticed an improvement, while of those receiving acupuncture, 44% noticed an improvement.

Find the P-value for the indicated hypothesis test. -An airline claims that the no-show rate for passengers booked on its flights is less than 6%. Of 380 randomly selected reservations, 18 were no-shows. Find the P-value for a test of the airline's claim.

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -When 12 bolts are tested for hardness, their indexes have a standard deviation of 41.7. Test the claim that the standard deviation of the hardness indexes for all such bolts is greater than 30.0. Use a 0.025 level of significance.

Complete the table to compare the z and t distributions. distribution distribution Shape Mean value Standard deviation value Requirements

Find the critical value or values of x $x ^ { 2 }$ based on the given information. - :\sigma<0.14 =23 \alpha=010

Determine whether the given conditions justify testing a claim about a population mean µ -The sample size is $\mathrm { n } = 25 , \sigma = 5.93$ , and the original population is normally distributed.

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion. -Use a significance level of $\alpha = 0.01$ to test the claim that $\mu > 2.85$ . The sample data consists of 9 scores for which $\bar { x }$ $= 3.25$ and $\mathrm { s } = 0.53$ .

An entomologist writes an article in a scientific journal which claims that fewer than 19 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.

Express the null hypothesis $\mathrm { H } _ { 0 }$ and the alternative hypothesis $\mathrm { H } _ { 1 }$ in symbolic form. Use the correct symbol ( $\mu , \mathrm { p }$ , $\sigma$ )for the indicated parameter. -An entomologist writes an article in a scientific journal which claims that fewer than 11 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Use the parameter p, the true proportion of fireflies unable to produce light.

Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test -A psychologist claims that more than 3 percent of the population suffers from professional problems due to extreme shyness. Identify the type II error for the test.

Define P-values. Explain the two methods of interpreting P-values.

In a hypothesis test regarding a population mean, the probability of a type II error, $\beta ,$ , depends on the true value of the population mean.

Assume that the data has a normal distribution and the number of observations is greater than fifty. Find the critical z value used to test a null hypothesis. -A researcher claims that the amounts of acetaminophen in a certain brand of cold tablets have a standard deviation different from the $\sigma = 3.3 \mathrm { mg }$ claimed by the manufacturer.

Express the null hypothesis $\mathrm { H } _ { 0 }$ and the alternative hypothesis $\mathrm { H } _ { 1 }$ in symbolic form. Use the correct symbol ( $\mu , \mathrm { p }$ , $\sigma$ )for the indicated parameter. -The manufacturer of a refrigerator system for beer kegs produces refrigerators that are supposed to maintain a true mean temperature, $\mu$ , of $48 ^ { \circ } \mathrm { F }$ , ideal for a certain type of German pilsner. The owner of the brewery does not agree with the refrigerator manufacturer, and claims he can prove that the true mean temperature is incorrect.