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Chapter 8: Q43E (page 452)

Consider making an inference about p1-p2, where there are x1successes inn1binomial trials and x2successes inn2binomial trials.

a. Describe the distributions of x1and x2.

b. Explain why the Central Limit Theorem is important in finding an approximate distribution forp^1-p^2

Short Answer

Expert verified

b. The central limit theorem approximates the distribution without considering the population's original distribution.

Step by step solution

01

Given information

We have, for the first trial

The number of binomial trials is n1and x1there are successes

For the second trial

There are binomial trials n2, and there arex2 successes.

02

Definition of binomial trials

When we have a prefixed number of trials to get the desired number of successes with a fixed probability of success throughout the experiment, the process will be well executed by binomial probabilities.

03

Importance of the central limit theorem

The central limit theorem helps to find the approximate distribution for the sample mean when the sample size is large enough (at least 30) regardless of the original distribution.

Hereand can be viewed as means of the number of successes per trial in the respective samples.

Hence the distribution of can be approximated using the central limit theorem, as it represents the difference in the sample mean for the number of successes per trial.

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Most popular questions from this chapter

Question:Quality control. Refer to Exercise 5.68. The mean diameter of the bearings produced by the machine is supposed to be .5 inch. The company decides to use the sample mean from Exercise 5.68 to decide whether the process is in control (i.e., whether it is producing bearings with a mean diameter of .5 inch). The machine will be considered out of control if the mean of the sample of n = 25 diameters is less than .4994 inch or larger than .5006 inch. If the true mean diameter of the bearings produced by the machine is .501 inch, what is the approximate probability that the test will imply that the process is out of control?

Question: Performance ratings of government agencies. The U.S. Office of Management and Budget (OMB) requires government agencies to produce annual performance and accounting reports (PARS) each year. A research team at George Mason University evaluated the quality of the PARS for 24 government agencies (The Public Manager, Summer 2008), where evaluation scores ranged from 12 (lowest) to 60 (highest). The accompanying file contains evaluation scores for all 24 agencies for two consecutive years. (See Exercise 2.131, p. 132.) Data for a random sample of five of these agencies are shown in the accompanying table. Suppose you want to conduct a paired difference test to determine whether the true mean evaluation score of government agencies in year 2 exceeds the true mean evaluation score in year 1.

Source: J. Ellig and H. Wray, 鈥淢easuring Performance Reporting Quality,鈥� The Public Manager, Vol. 37, No. 2, Summer 2008 (p. 66). Copyright 漏 2008 by Jerry Ellig. Used by permission of Jerry Ellig.

a. Explain why the data should be analyzedusing a paired difference test.

b. Compute the difference between the year 2 score and the year 1 score for each sampled agency.

c. Find the mean and standard deviation of the differences, part

b. Use the summary statistics, part c, to find the test statistic.

e. Give the rejection region for the test using a = .10.

f. Make the appropriate conclusion in the words of the problem.

Is honey a cough remedy? Refer to the Archives of Pediatrics and Adolescent Medicine (Dec. 2007) study of honey as a children鈥檚 cough remedy, Exercise 8.23 (p. 470). The data (cough improvement scores) for the 33 children in the DM dosage group and the 35 children in the honey dosage group are reproduced in the table below. In Exercise 8.23, you used a comparison of two means to determine whether 鈥渉oney may be a preferable treatment for the cough and sleep difficulty associated with childhood upper respiratory tract infection.鈥� The researchers also want to know if the variability in coughing improvement scores differs for the two groups. Conduct the appropriate analysis, using 伪=0.10

4.135 Suppose xhas an exponential distribution with 胃=1. Find

the following probabilities:

a.P(x>1)b.P(x鈮�3)cP(x>1.5)d.P(x鈮�5)

Question: Forecasting daily admission of a water park. To determine whether extra personnel are needed for the day, the owners of a water adventure park would like to find a model that would allow them to predict the day鈥檚 attendance each morning before opening based on the day of the week and weather conditions. The model is of the form

where,

y = Daily admission

x1 = 1 if weekend

0 otherwise

X2 = 1 if sunny

0 if overcast

X3 = predicted daily high temperature (掳F)

These data were recorded for a random sample of 30 days, and a regression model was fitted to the data.

The least squares analysis produced the following results:

with

  1. Interpret the estimated model coefficients.
  2. Is there sufficient evidence to conclude that this model is useful for predicting daily attendance? Use 伪 = .05.
  3. Is there sufficient evidence to conclude that the mean attendance increases on weekends? Use 伪 = .10.
  4. Use the model to predict the attendance on a sunny weekday with a predicted high temperature of 95掳F.
  5. Suppose the 90% prediction interval for part d is (645, 1,245). Interpret this interval.
See all solutions

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