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Chapter 7: Q120SE (page 442)

Latex allergy in health care workers. Refer to the Current Allergy & Clinical Immunology (March 2004) study of n = 46 hospital employees who were diagnosed with a latex allergy from exposure to the powder on latex gloves, Exercise 6.112 (p. 375). The number of latex gloves used per week by the sampled workers is summarized as follows: \(\bar x = 19.3\) and s = 11.9. Let \(\mu \) represent the mean number of latex gloves used per week by all hospital employees. Consider testing \({H_0}:\mu = 20\) against \({H_a}:\mu < 20.\)

a. Give the rejection region for the test at a significance level of \(\alpha = 0.01.\)

Short Answer

Expert verified

a. The rejection region is \({Z_c} < - 2.33.\)

Step by step solution

01

Given Information

The number of hospital employees who were diagnosed with a latex allergy, n=46.

The sample mean is, \(\bar x = 19.3.\)

The sample standard deviation is,\(s = 11.9.\)

The test hypothesis is \({H_0}:\mu = 20\) against \({H_a}:\mu < 20.\)

02

State the large sample one-tailed test of hypothesis about a population mean.

The elements of the one-tailed hypothesis test are the null hypothesis\(\left( {{H_0}} \right)\), alternative hypothesis\(\left( {{H_a}} \right)\), and test statistic. The sample statistic is used to decide whether to reject the null hypothesis.

A one-tailed test of a hypothesis is one in which the alternative hypothesis is directional and includes the symbol 鈥�<鈥� or 鈥�>鈥�.

The one-tailed hypothesis are:

  • \({H_0}:\mu = {\mu _0}\)against \({H_a}:\mu > {\mu _0}\)
  • \({H_0}:\mu = {\mu _0}\) against \({H_a}:\mu < {\mu _0}\)
03

Compute the rejection region for the test at a significance level \(\alpha  = 0.01.\)

a.

Let \(\mu \)represent the mean number of latex gloves used per week by all hospital employees.

The null and alternative hypothesis are:

\(\begin{aligned}{l}{H_0}:\mu = 20\\{H_a}:\mu < 20\end{aligned}\)

The significance level, \(\alpha = 0.01.\)

The rejection region for the one-tailed test is \({Z_c} < - {Z_\alpha }\).

The \( - {Z_\alpha }\) corresponding significance level \(\alpha = 0.01\) from the standard normal table is -2.33.

Therefore,

The rejection region can written as:

\({Z_c} < - {Z_\alpha } \Rightarrow {Z_c} < - 2.33.\)

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Most popular questions from this chapter

For each of the following situations, determine the p-value and make the appropriate conclusion.

a.\({H_0}:\mu \le 25\),\({H_a}:\mu > 25\),\(\alpha = 0.01\),\(z = 2.02\)

b.\({H_0}:\mu \ge 6\),\({H_a}:\mu < 6\),\(\alpha = 0.05\),\(z = - 1.78\)

c.\({H_0}:\mu = 110\),\({H_a}:\mu \ne 110\),\(\alpha = 0.1\),\(z = - 1.93\)

d. \({H_0}:\mu = 10\), \({H_a}:\mu \ne 10\), \(\alpha = 0.05\), \(z = 1.96\)

If you select a very small value for 伪when conducting a hypothesis test, will 尾 tend to be big or small? Explain.

Factors that inhibit learning in marketing. What factors inhibit the learning process in the classroom? To answer this question, researchers at Murray State University surveyed 40 students from a senior-level marketing class (Marketing Education Review). Each student was given a list of factors and asked to rate the extent to which each factor inhibited the learning process in courses offered in their department. A 7-point rating scale was used, where 1 = 鈥渘ot at all鈥� and 7 = 鈥渢o a great extent.鈥� The factor with the highest rating was instructor related: 鈥淧rofessors who place too much emphasis on a single right answer rather than overall thinking and creative ideas.鈥� Summary statistics for the student ratings of this factor are\(\overline x = 4.70\),\(s = 1.62\)

a. Conduct a test to determine if the true mean rating for this instructor-related factor exceeds 4. Use\(\alpha = 0.05\).Interpret the test results.

b. Examine the results of the study from a practical view, and then discuss why 鈥渟tatistically significant鈥� does not always imply 鈥減ractically significant.鈥�

c. Because the variable of interest, rating, is measured on a 7-point scale, it is unlikely that the population of ratings will be normally distributed. Consequently, some analysts may perceive the test, part a, to be invalid and search for alternative methods of analysis. Defend or refute this argument

Cooling method for gas turbines. During periods of high electricity demand, especially during the hot summer months, the power output from a gas turbine engine can drop dramatically. One way to counter this drop in power is by cooling the inlet air to the gas turbine. An increasingly popular cooling method uses high-pressure inlet fogging. The performance of a sample of 67 gas turbines augmented with high-pressure inlet fogging was investigated in the Journal of Engineering for Gas Turbines and Power (January 2005). One performance measure is heat rate (kilojoules per kilowatt per hour). Heat rates for the 67 gas turbines are listed in the table below. Suppose that standard gas turbines have heat rates with a standard deviation of 1,500 kJ/kWh. Is there sufficient evidence to indicate that the heat rates of the augmented gas turbine engine are more variable than the heat rates of the standard gas turbine engine? Test using a = .05.

A t-test is conducted for the null hypothesis H0:渭=10versus the alternative Ha:渭>10for a random sample of n = 17 observations. The test results are t = 1.174, p-value = 0.1288.

a. Interpret the p-value.

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