Exam 8: Hypothesis Testing

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. -A manufacturer considers his production process to be out of control when defects exceed 3%. In a random sample of 85 items, the defect rate is 5.9% but the manager claims that this is only a sample fluctuation and production is not really out of control. At the 0.01 level of significance, test the manager's claim.

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. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is failure to reject the null hypothesis, state the conclusion in nontechnical terms.

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 $\mathrm { z }$ is the critical value. To find the lower critical value, the negat 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 $\mathrm { n } = 144$ and $\alpha = 0.10$ .

The manufacturer of a refrigerator system for beer kegs produces refrigerators that are supposed to maintain a true mean temperature, $\mu$ , of $45 ^ { \circ } \mathrm { F }$ , 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. 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.

Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test -Carter Motor Company claims that its new sedan, the Libra, will average better than 23 miles per gallon in the city. Identify the type I error for the test.

Provide an appropriate response. -When testing hypotheses about a mean, the decision must be made as to the distribution to be used. Discuss the decision process used to decide whether z or t or neither is the proper distribution.

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 company claims that the proportion of defectives among a particular model of computers is 4%. In a shipment of 200 such computers, there are 10 defectives.

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. -A cereal company claims that the mean weight of the cereal in its packets is at least 14 oz. The weights (in ounces)of the cereal in a random sample of 8 of its cereal packets are listed below. 14.6 13.8 14.1 13.7 14.0 14.4 13.6 14.2 At the 0.01 significance level, test the claim that the mean weight is 14 ounces.

Suppose that you perform a hypothesis test regarding a population mean, and that the evidence does not warrant rejection of the null hypothesis. When formulating the conclusion to the test, why is the phrase "fail to reject the null hypothesis" more accurate than the phrase "accept the null hypothesis"?

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -Systolic blood pressure levels for men have a standard deviation of 19.7 mm Hg. A random sample of 31 women resulted in blood pressure levels with a standard deviation of 23.2 mm Hg. Use a 0.05 significance level to test the claim that blood pressure levels for women have the same variation as those for men.

A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO, p, is less than 3 in every ten thousand. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is failure to reject the null hypothesis, state the conclusion in nontechnical terms.

Find the P-value for the indicated hypothesis test. -In a sample of 88 children selected randomly from one town, it is found that 8 of them suffer from asthma. Find the P-value for a test of the claim that the proportion of all children in the town who suffer from asthma is equal to 11%.

Find the P-value for the indicated hypothesis test. -In a sample of 47 adults selected randomly from one town, it is found that 9 of them have been exposed to a particular strain of the flu. Find the P-value for a test of the claim that the proportion of all adults in the town that have been exposed to this strain of the flu is 8%.

Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test -The owner of a football team claims that the average attendance at games is over 60,000, and he is therefore justified in moving the team to a city with a larger stadium. Identify the type II error for the test.

Use the given information to find the P-value -The test statistic in a right-tailed test is z = 0.52.

Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test -An entomologist writes an article in a scientific journal which claims that fewer than 16 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Identify the type I error for the test.

Suppose that you wish to use a hypothesis test to test a claim made by a juice bottling company regarding the mean amount of juice in its 16 oz bottles. Why does the original claim sometimes become the null hypothesis, and why does it sometimes become the alternative hypothesis? Give an example of a claim which would become the null hypothesis and an example of a claim would become the alternative hypothesis.

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected. -For randomly selected adults, IQ scores are normally distributed with a standard deviation of 15 . The scores of 1 . randomly selected college students are listed below. Use a $0.10$ significance level to test the claim that the standa deviation of $\mathrm { IQ }$ scores of college students is less than 15. 115 128 107 109 116 124 135 127 115 104 118 126 129 133

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. Last year an appliance manufacturer received many complaints about the high rate of defects among its washing machines. Approximately 9% of the machines were defective in some way. This year the company tightened up its quality control procedures. The latest shipment of 250 washing machines contained 2 defectives.

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. -Various temperature measurements are recorded at different times for a particular city. The mean of 20°C is obtained for 40 temperatures on 40 different days. Assuming that = 1.5°C, test the claim that the population mean is 23°C. Use a 0.05 significance level.