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Chapter 7: Q94E (page 431)

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.

Short Answer

Expert verified

There is insufficient information to infer that the mean heat rates of a gas turbine with pressure inlet fogging exceeds 1,500 KJ/kWh when the alpha is 0.05.

Step by step solution

01

The variable is the Gas turbine

There is insufficient information to infer that the mean heat rates of a gas turbine with pressure inlet fogging exceeds 1,500 KJ/kWh when the alpha is 0.05.

We may use the following direct command for t testing:n = 61mu=1500##populationmean##totest:Ho:mu=1,500vsH1:mu>1500t.test(GASTURBINE,mu=1500,alternative="greater",conf.level=0.05,paired=FALSE)
02

One-Sample t-test

Data: GAS TURBINEt = alternative hypothesis: true mean is greater than 15005 percent confidence interval:11478.69InfSample estimates:Mean of x = 11132.93

If the p-value is smaller than the alpha value, we reject Ho. We reject Ho.

There is insufficient information to infer that the mean heat rates of a gas turbine with pressure inlet fogging exceeds 1,500 KJ/kWh when the alpha is 0.05.

The type 1 and type 2 error:

We reject Ho when it is true is a type 1 error here; we reject Ho that is type error occurs, and a type 2 error is the mean hears rates of a gas turbine with pressure inlet fogging exceeding 1,500 KJ/kWh when alpha is 0.05.

The fail to reject Ho.

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Most popular questions from this chapter

Trading skills of institutional investors. The trading skills of institutional stock investors were quantified and analyzed in The Journal of Finance (April 2011). The study focused on 鈥渞ound-trip鈥� trades, i.e., trades in which the same stock was both bought and sold in the same quarter. Consider a random sample of 200 round-trip trades made by institutional investors. Suppose the sample mean rate of return is 2.95% and the sample standard deviation is 8.82%. If the true mean rate of return of round-trip trades is positive, then the population of institutional investors is considered to have performed successfully.

a. Specify the null and alternative hypotheses for determining whether the population of institutional investors performed successfully.

b. Find the rejection region for the test using\(\alpha = 0.05\).

c. Interpret the value of\(\alpha \)in the words of the problem.

d. A Minitab printout of the analysis is shown below. Locate the test statistic and p-value on the printout. (Note: For large samples, z 鈮� t.)

e. Give the appropriate conclusion in the words of the problem.

For each of the following rejection regions, sketch the sampling distribution for z and indicate the location of the rejection region.

a. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.1\)

b. \({H_0}:\mu \le {\mu _0}\) and \({H_a}:\mu > {\mu _0};\alpha = 0.05\)

c. \({H_0}:\mu \ge {\mu _0}\) and \({H_a}:\mu < {\mu _0};\alpha = 0.01\)

d. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.05\)

e. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.1\)

f. \({H_0}:\mu = {\mu _0}\) and \({H_a}:\mu \ne {\mu _0};\alpha = 0.01\)

g. For each rejection region specified in parts a鈥揻, state the probability notation in z and its respective Type I error value.

Packaging of a children鈥檚 health food. Can packaging of a healthy food product influence children鈥檚 desire to consume the product? This was the question of interest in an article published in the Journal of Consumer Behaviour (Vol. 10, 2011). A fictitious brand of a healthy food product鈥攕liced apples鈥攚as packaged to appeal to children (a smiling cartoon apple was on the front of the package). The researchers showed the packaging to a sample of 408 school children and asked each whether he or she was willing to eat the product. Willingness to eat was measured on a 5-point scale, with 1 = 鈥渘ot willing at all鈥� and 5 = 鈥渧ery willing.鈥� The data are summarized as follows: \(\bar x = 3.69\) , s = 2.44. Suppose the researchers knew that the mean willingness to eat an actual brand of sliced apples (which is not packaged for children) is \(\mu = 3\).

a. Conduct a test to determine whether the true mean willingness to eat the brand of sliced apples packaged for children exceeded 3. Use\(\alpha = 0.05\)

to make your conclusion.

b. The data (willingness to eat values) are not normally distributed. How does this impact (if at all) the validity of your conclusion in part a? Explain.

Which hypothesis, the null or the alternative, is the status-quo hypothesis? Which is the research hypothesis?

Consider the test \({H_0}:\mu = 70\) versus \({H_a}:\mu \ne 70\) using a large sample of size n = 400. Assume\(\sigma = 20\).

a. Describe the sampling distribution of\(\bar x\).

b. Find the value of the test statistic if\(\bar x = 72.5\).

c. Refer to part b. Find the p-value of the test.

d. Find the rejection region of the test for\(\alpha = 0.01\).

e. Refer to parts c and d. Use the p-value approach to

make the appropriate conclusion.

f. Repeat part e, but use the rejection region approach.

g. Do the conclusions, parts e and f, agree?
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