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CHAPTER 7. Homework:5,7,9,11,17,22,23,25,29,33,37,41,45,51,59,65,77,79 <Example 7.1>:

CHAPTER 7

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Homework:5,7,9,11,17,22,23,25,29,33,37,41,45,51,59,65,77,79

<Example 7.1>:

The U.S. Bureau of Census publishes annual price figures for new mobile homes in Construction Reports. The information is obtained from sampling, not from a census. Suppose a random sample of 40 new mobile homes yields the prices shown in Table 7.1. Data are in thousands of dollars, rounded to the nearest hundred.

Table 7.1

8.4 9.9 13.9 13.9 14.1 15.8 16.6 16.8

17.0 17.6 18.8 20.0 21.4 22.1 22.4 22.8

22.8 24.0 24.4 24.7 24.8 26.2 26.4 26.7

26.8 27.0 28.4 29.3 29.3 29.5 30.6 31.1

31.4 31.9 32.2 32.4 33.0 33.5 35.1 37.2

(a) Find the sample mean and sample standard

deviation.

(b) Find a point estimator for the population mean and a point estimator for the population standard deviation.

Output 7.01: Prices of Mobile Homes

SAS summary statistics

Analysis Variable : PRICE

N Mean Std Dev Minimum Maximum

-----------------------------------------

40 24.26 7.20 8.40 37.20

-----------------------------------------

Figure 7.1 Prices of Mobil Homes

35

30

25

20

15

10

<DISCUSSION>:

(1) Point estimators do not provide any information of the reliability for itself. An interval estimator can provide the reliability information. Therefore, we want to construct an interval estimator of the population mean.

(2) Usually, we need to have several reasonable assumptions about the population before we can actually construct the interval estimator.

Interval Estimator An interval estimator is a rule that tells us how to use sample data to form an interval estimator of the population parameters.

Confidence Coefficient:

The confidence coefficient of an interval estimator is the probability that this interval estimator encloses the population parameter.

Confidence Level:

The confidence level is the confidence coefficient expressed as a percentage.

(a) Is the sample representative?

We need to make sure that the sample is randomly selected from the population. This is one of the most basic principles of statistical inference.

(b) Check the sample size

If the sample size is greater than 30, you can use the large sample procedure because that:

(i) the central limit theorem ensures at least approximate normality of the distribution of the sample mean and

(ii) the law of large number ensures that the sample standard deviation provides a good estimator of the population standard deviation.

(c) Find the sample mean and standard error of the sample mean.

(d) Find za/2, where a is equal to one minus confidence coefficient.

The (1-)100% confidence interval for the population mean is then given by

Interpretation of a confidence interval:

For any single given confidence interval of the

population mean, we don't know whether it contains

the population mean or not. But if this confidence

interval (construction) procedure is used on a large

number of random samples, then about (1 - a ) * 100%

of the intervals do contain

the unknown population mean.

<Example 7.2> (Basic)

Find a 90% confidence interval for the population mean, if

(a). n = 125, sample mean = 13.1, and sample variance = 0.086.

(b). n = 50, sample mean = 21.9, and sample variance = 3.44.

<Example 7.3> (Basic)

Acid rain, caused by the reaction of certain air pollutants with rain water, appears to be a growing problem in the north eastern section of the United States. Pure rainfall through clean air registers a pH value of 5.7. Suppose that water samples from 40 rainfalls are analyzed for pH and that mean and standard deviation are equal to 3.7 and 0.5, respectively. Find a 99% C.I. for the average pH in rainfall and interpret this interval. What assumptions must be made for the confidence interval to be valid?

<Example 7.4> (Intermediate)

Suppose that a random sample (with sample mean ) of size n >= 30 is to be taken from a population with mean m and standard deviation s.

(a) Determine the probability that the interval

will contain the population mean.

(b) Interpret your result in part (a) in terms of percentages.

<Example 7.5> (Advance)

(a) Figure 7.3 is the box plot of the density of earth data set.

Table 1 Measurements of the density of the earth

5.505.614.884.075.265.55

5.365.295.585.655.575.53

5.625.295.445.345.795.10

5.275.395.425.475.345.46

5.305.755.865.63

(b) Figure 7.5 is the bax plots of the same data wothout one extreme value. Briefly discuss the main features such as symmetry and outlier issue of this distribution .

(c) What is your estimate of the density of the earth based on this density data set?

Output 7.02: SAS summary statistics

Analysis Variable: DENSITY (All)

N Mean Std Dev Minimum Maximum

----------------------------------------

28 5.40 0.33 4.07 5.86

----------------------------------------

Analysis Variable: DENSITY (without 4.07)

N Mean Std Dev Minimum Maximum

-----------------------------------------

27 5.45 0.21 4.88 5.86

-----------------------------------------

Figure 7.3 Density of the Earth

5.5

5.0

4.5

4.0

Figure 7.5 Density of the Earth

5.8

5.6

5.4

5.2

5.0

The student t Distribution

(a) The p.d.f. of a t random variable is given by

(b) The limiting distribution (as n approaches

infinity) of t is normal distribution.

(c) Student t distribution is symmetric about itsmedian or mean.

(d) The variance of a student t random variable is larger than the variance of a standard normal random variable.

Figure 7.7 Student t Distribution

0.4

Normal Curve

0.3

0.2

t curve with 4 d.f.

0.1

0.0

0

2

4

-4

-2

Quantile

aa

(2). Steps to construct the confidence interval:

(a) Is this sample representative?

We need to make sure that the sample is randomly selected from the population.

(b) Is the population approximately normal?

Note that this small sample confidence interval is applicable only if the population has a normal distribution!!! This can be assessed roughly by

using graphical tools to display the data.

(c) Find the sample mean and standard error

(d) Find ta/2,n-1 from Table VI, where (1 - a) is the confidence coefficient and n is the sample size.

(e) The (1-a)*100% confidence interval is

<Example 7.6> (Basic)

A very costly experiment has been conducted to evaluate a new process for producing synthetic diamonds. Six diamonds have been produced by the new process with recorded weight 0.46, 0.61, 0.52, 0.48, 0.57, and 0.54 karat.

(a) What assumptions do we need to use the t confidence interval?

(b) Find a 95% C.I. of the population mean.

Output 7.03: SAS summary statistics

Analysis Variable: DIAMOND

N Mean Std Dev Minimum Maximum

----------------------------------------

6 0.53 0.056 0.46 0.61

----------------------------------------

<Example 7.7> (Basic)

A random sample of size n=12 was selected from a normally distributed population. The sample mean is 47.1.

(a) Find the 95% C.I. of the population mean if the sample variance is 4.7.

(b) Find the 95% C.I. of the population mean if the sample size is 64 and sample variance is 4.7.

(c) Interpret the intervals in (a) and (b).

<Example 7.8> (Advance)

It is recognized that the cigarette smoking has deleterious effect on lung function. A recent study found that the carbon monoxide diffusing capacity (DC) of the lung for 20 current smokers are as follows:

61.675 71.210 73.003 73.154 76.014

82.115 84.023 88.017 88.602 89.222

90.479 90.677 91.052 92.295 100.615

102.754 103.768 106.755 108.579 123.086

Output 7.04: SAS summary statistics

Analysis Variable: CARBON

N Mean Std Dev Minimum Maximum

-----------------------------------------

20 89.85 14.90 61.68 123.09

-----------------------------------------

(a). Find the sample mean and sample standard

deviation.

(b). What assumptions do you need to construct the t confidence interval of DC for current smokers.

(c). Use stem-and-leaf display to check these assumptions.

(d). Find 95% C.I. of DC for current smokers.

Figure 7.8 Diffusing Capacity

120

110

100

90

80

70

60

Quantiles

Properties of Sample Proportion

(1) Sample proportion is = x / n.

(2) The expectation of sample proportion is

E( ) = p.

(3). Variance of is approximately

(4). For a sufficiently large sample, the sampling distribution of is approximately normal.

(5). Usually, the sample size is large enough if the interval does not include 0 or 1.

If the sample size is large enough, then an approximate 1- confidence interval for p is given by

<Example 7.9> (Basic)

Suppose that 6841 U.S. households are selected at random in order to estimate the proportion, p, of all U.S. households that have a computer. If 2470 out of the 6841 households chosen have a computer, find the 95% C.I. for p.

<Example 7.10> (Intermediate)

A telephone survey conducted by the Florida State University's Policy Science Program found that 74% of the 983 responding Florida adults were in favor of raising the drinking age from 19 to 21.

(a). What assumptions is necessary for the 74% to be valid point estimator of the percentages of all the adult Floridians who favor raising the drinking age from 19 to 21?

(b). Is it possible that the assumption in part (a) might not be satisfied in a telephone survey?

(c). Assume that the assumptions in part (a) were satisfied by the pollsters, find a 95% C.I. for p.

(1). Width and Confidence Level:

The width of a confidence interval increased when the confidence level of this interval increased.

<Example 7.11> (Intermediate)

The U.S. Energy Information Administration surveys household in order to obtain data on monthly fuel expenditures for household vehicles. Suppose 60 monthly fuel expenditures for household vehicles are randomly selected and their mean is equal to 58.56 with a 20.65 standard deviation.

(a). Find a 95% confidence interval of household monthly expenditure on vehicles.

(b). Find a 90% confidence interval of household monthly expenditure on vehicles.

(c). Find a 99% confidence interval of household monthly expenditure on vehicles.

(d). Discuss the relationship between confidence level and width of confidence interval.

Error Bound of Confidence Interval

The error bound of the estimator of the population mean at confidence level 1 - a is

It is equal to (about) the half width

of the confidence interval

Sample Size Requirements for Estimating the Population Mean

The sample size required for a (1 - a)*100% confidence interval for the population mean with given error bound B is given by

<Example 7.12> (Intermediate)

Suppose that you wish to estimate the mean pH of rainfalls in an area that suffers heavy pollution due to the discharge of smoke from a power plant.

(a). Assume that you know that sample standard deviation is 0.5 pH and that you wish to estimate within 0.1 of population mean with probability of 0.95. Approximately how many rainfalls would have to include in your sample?

(b). Would it be valid to select all your specimens from a single rainfall?

(c). Suppose that sample s.d. is 0.2. Repeat part (a). Do you trust your answer? Explain.

The sample size required for a (1-a)*100% confidence interval for the population proportion p with a given error bound B is given by

<Example 7.13> (Advance)

Find the spfor the following proportion p for n=100.

(a). P=0.5.

(b). P=0.6.

(c). P=0.7.

(d). P=0.4.

(e). P=0.3.

Figure 7.10

•

•

•

•

•

0.045

0.045

•

•

Standard Error

0.035

0.035

•

•

0.025

0.025

•

•

0.5

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

Probabilities

<Example 7.14> (Intermediate)

Suppose that you want to estimate a binomial parameter p correct to within 0.04 (i.e., B = 0.04) at 95% confidence level.

(a). How large should the sample size n be?

(b). You suspect that P is equal to some value between 0.1 and 0.3. How large should the sample size n be?

(c). You suspect that P is equal to some value between 0.6 and 0.8. How large should the sample size n be?