Exam 8: Hypothesis Testing

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. A manufacturer uses a new production method to produce steel rods. A random sample of 17 steel rods resulted in lengths with a standard deviation of 4.7 cm. At the 0.10 significance level, test the claim that the new production method has lengths with a standard deviation different from 3.5 cm, which was the standard deviation for the old method.

Find the value of the test statistic $z$ using $z = \frac { \hat { p } - p } { \sqrt { \frac { p q } { n } } }$ . The claim is that the proportion of drowning deaths of children attributable to beaches is more than $0.25$ , and the sample statistics include $n = 696$ drowning deaths of children with $30 \%$ of them attributable to beaches.

Identify the null hypothesis, alternative hypothesis, test statistic, $P$ -value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. The health of employees is monitored by periodically weighing them in. A sample of 54 employees has a mean weight of $183.9 \mathrm { lb }$ . Assuming that $\sigma$ is known to be $121.2 \mathrm { lb }$ , use a $0.10$ significance level to test the claim that the population mean of all such employees weights is less than 200 lb.

Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim. Carter Motor Company claims that its new sedan, the Libra, will average better than 32 Miles per gallon in the city. 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.

A Type I error is the mistake of ________________ when it is actually true.

Provide an appropriate response. Tim believes that a coin is coming up tails less than $50 \%$ of the time. He tests the claim $p < 0.5$ . In 100 tosses, the coin comes up tails 57 times. What is the value of the sample proportion? Do you think the $P$ -value will be small or large and what should Tim conclude about the claim $p < 0.5$ ?

Sam wanted to test a claim about the mean of a population whose standard deviation was unknown. He picked a simple random sample of size 20 from the population. Lou wanted to test a claim about a mean of a different population whose standard deviation was known. He picked a simple random sample of size 22 from that population. George said that Sam would need to determine whether his sample was from a normally distributed population because the population standard deviation was unknown. He said that Lou would not need to do this since for his test the population standard deviation was known. Is George right?

Solve the problem. For large numbers of degrees of freedom, the critical $\chi ^ { 2 }$ values can be approximated as follows: $\chi ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$ , where $\mathrm { k }$ is the number of degrees of freedom and $z$ is the critical value. To find the lower critical value, the negative $z$ -value is used, to find the upper critical value, the positive $z$ -value is used. Use this approximation to estimate the critical value of $\chi ^ { 2 }$ in a two-tailed hypothesis test with $n = 104$ and $\alpha = 0.10$ .

Which of the following is a requirement for testing a claim about a population proportion?

Solve the problem. What do you conclude about the claim below? Do not use formal procedures or exact calculations. Use only the rare event rule and make a subjective estimate to determine whether the event is likely. Claim: A roulette wheel is fair and in 40 consecutive spins of the wheel, black shows up 23 times. (A roulette wheel has 38 equally likely slots of which 18 are black).

A _____________ error is the mistake of rejecting the null hypothesis when it is actually true.

Find the critical value or values of $\chi ^ { 2 }$ based on the given information. :\sigma<0.14 n=23 \alpha=0.10

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. Test the claim that for the adult population of one town, the mean annual salary is given by $\mu = \$ 30,000$ . Sample data are summarized as $n = 17 , \bar { x } = \$ 22,298$ , and $s = \$ 14,200$ . Use a significance level of $\alpha = 0.05$ .

Find the critical value or values of $\chi ^ { 2 }$ based on the given information. :\sigma<0.629 n=19 \alpha=0.025

Solve the problem. Suppose that you are conducting a study on the effectiveness of a new teaching method and that you wish to use a hypothesis test to support your claim regarding the mean test score under this method. What restrictions are there in the wording of the claim? Will your claim become the null hypothesis or the alternative hypothesis, or does it depend on the situation? Give an example of a claim which is incorrectly worded.

Test the given claim. Use the P-value method or the traditional method as indicated. the null hypothesis, alternative hypothesis, test statistic, critical value(s)or P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. The mean resting pulse rate for men is 72 beats per minute. A simple random sample of men who regularly work out at Mitch's Gym is obtained and their resting pulse rates (in beats per minute)are listed below. Use a 0.05 significance level to test the claim that these sample pulse rates come from a population with a mean less than 72 beats per minute. Assume that the standard deviation of the resting pulse rates of all men who work out at Mitch's Gym is known to be 6.6 beats per minute. Use the traditional method of testing hypotheses. 54 59 69 84 74 64 69 70 66 80 59 71 76 63

Express the null hypothesis and the alternative hypothesis in symbolic form. Use the correct symbol $( \mu , p , \sigma )$ 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.

Provideanappropriateresponse.Completethefollowingtableonhypothesistesting. Test about Distribution Assumptions Mean Median Proportion Variance

Solve the problem. Use the P-value method to test the claim that the population standard deviation of the systolic blood pressures of adults aged 40-50 is equal to 22 mmHg. The sample statistics are as follows: nx==23, 132.2 mmHg, s = 26.6 mmHg. Be sure to state the hypotheses, the value of this test statistic, the P-value, and your conclusion. Use a significance level of 0.05.

Express the original claim in symbolic form. Claim: $60 \%$ of homes have smoke detectors.