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Chapter 7: Q111S (page 441)
If you select a very small value for when conducting a hypothesis test, will tend to be big or small? Explain.
The lower values of significance level increase the type II error .
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In a test of \({H_0}:\mu = 100\) against \({H_a}:\mu \ne 100\), the sample data yielded the test statistic z = 2.17. Find the p-value for the test.
Authorizing computer users with palm prints. Access to computers, email, and Facebook accounts is achieved via a password鈥攁 collection of symbols (usually letters and numbers) selected by the user. One problem with passwords is that persistent hackers can create programs that enter millions of combinations of symbols into a target system until the correct password is found. An article in IEEE Pervasive Computing (October-December 2007) investigated the effectiveness of using palm prints to identify authorized users. For example, a system developed by Palmguard, Inc. tests the hypothesis
\({H_0}\): The proposed user is authorized
\({H_a}\): The proposed user is unauthorized
by checking characteristics of the proposed user鈥檚 palm print against those stored in the authorized users鈥� data bank.
a. Define a Type I error and Type II error for this test. Which is the more serious error? Why?
Question: Water distillation with solar energy. In countries with a water shortage, converting salt water to potable water is big business. The standard method of water distillation is with a single-slope solar still. Several enhanced solar energy water distillation systems were investigated in Applied Solar Energy (Vol. 46, 2010). One new system employs a sun-tracking meter and a step-wise basin. The new system was tested over 3 randomly selected days at a location in Amman, Jordan. The daily amounts of distilled water collected by the new system over the 3 days were 5.07, 5.45, and 5.21 litres per square meter ( l / m2 ). Suppose it is known that the mean daily amount of distilled water collected by the standard method at the same location in Jordan is 碌 = 1.4 / I m2.
a. Set up the null and alternative hypotheses for determining whether the mean daily amount of distilled water collected by the new system is greater than 1.4.
b. For this test, give a practical interpretation of the value 伪 = 0.10.
c. Find the mean and standard deviation of the distilled water amounts for the sample of 3 days.
d. Use the information from part c to calculate the test statistic.
e. Find the observed significance level (p-value) of the test.
f. State, practically, the appropriate conclusion.
Producer's and consumer's risk. In quality-control applications of hypothesis testing, the null and alternative hypotheses are frequently specified as\({H_0}\)The production process is performing satisfactorily. \({H_a}\): The process is performing in an unsatisfactory manner. Accordingly, \(\alpha \) is sometimes referred to as the producer's risk, while \(\beta \)is called the consumer's risk (Stevenson, Operations Management, 2014). An injection molder produces plastic golf tees. The process is designed to produce tees with a mean weight of .250 ounce. To investigate whether the injection molder is operating satisfactorily 40 tees were randomly sampled from the last hour's production. Their weights (in ounces) are listed in the following table.
a. Write \({H_0}\) and \({H_a}\) in terms of the true mean weight of the golf tees, \(\mu \).
b. Access the data and find \(\overline x \)and s.
c. Calculate the test statistic.
d. Find the p-value for the test.
e. Locate the rejection region for the test using\({H_a} = 0.01\).
f. Do the data provide sufficient evidence to conclude that the process is not operating satisfactorily?
g. In the context of this problem, explain why it makes sense to call \(\alpha \)the producer's risk and \(\beta \)the consumer's risk.
FDA certification of new drugs. Pharmaceutical companies spend billions of dollars per year on research and development of new drugs. The pharmaceutical company must subject each new drug to lengthy and involved testing before receiving the necessary permission from the Food and Drug Administration (FDA) to market the drug. The FDA鈥檚 policy is that the pharmaceutical company must provide substantial evidence that a new drug is safe prior to receiving FDA approval, so that the FDA can confidently certify the safety of the drug to potential consumers.
a. If the new drug testing were to be placed in a test of hypothesis framework, would the null hypothesis be that the drug is safe or unsafe? The alternative hypothesis?
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