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ROHANA BINTI ABDUL HAMID

INSTITUTE FOR ENGINEERING MATHEMATICS (IMK)

UNIVERSITI MALAYSIA PERLIS

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CHAPTER 4 (PART 1)

STATISTICAL INFERENCES

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4.1 INTRODUCTIONThe field of statistical inference consist of those

methods used to make decisions or to draw conclusions about a population.

These methods utilize the information contained in a sample from the population in drawing conclusions.

Statistical Inference may be divided into two major areas:

a)parameter estimation and

b) hypothesis testing.

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Definition 4.1: An Interval Estimate

In interval estimation, an interval is constructed around the point estimate and it is stated that this interval is likely to contain the corresponding population parameter.

4.2 PARAMETER ESTIMATION

Definition 4.2: Confidence Level and Confidence Interval

Each interval is constructed with regard to a given confidence level and is called a confidence interval.

The confidence level associated with a confidence interval states how much confidence we have that this interval contains the true population parameter.

The confidence level is denoted by

1 100%

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4.2.1 Confidence Interval Estimates for Population Mean

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EXAMPLE 4.1

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SOLUTION

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96.1025.0

2

ZZ

2

Z

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EXAMPLE 4.2

A publishing company has just published a new textbook. Before the company decides the price at which to sell this textbook, it wants to know the average price of all such textbooks in the market. The research department at the company took a sample of 36 comparable textbooks and collected the information on their prices. This information produced a mean price RM 70.50 for this sample. It is known that the standard deviation of the prices of all such textbooks is RM4.50.

Construct a 90% confidence interval for the mean price of all such college textbooks.

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SOLUTION

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1.09.01

65.105.0

2

1.0

2

ZZZ

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4.2.2 The (1 )100% Confidence Interval Estimates

on the Differences between Two Population Means,

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16

17

021 ( there is no difference between the two population mean)

If the confidence interval includes 0 we can say that there is no significant difference between the means of the two populations, at a given level of confidence.

EXAMPLE 4.3

The scientist wondered whether there was a difference in the average daily intakes of dairy products between men and women. He took a sample of n=50 adult women and recorded their daily intakes of dairy products in grams per day.

He did the same for adult men. A summary of his sample results is listed below.

Construct a 95% confidence interval for the difference in the average daily intakes of daily products for men and women. Can you conclude that there is a difference in the average daily intakes of daily products for men and women?

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Men Women

Sample size 50 50

Sample mean 780 grams per day 762 grams per day

Sample standard deviation

35 30

SOLUTION

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2

22

1

21

2

21 n

s

n

szxx

50

30

50

3596.1762780

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78.1218

]78.30,22.5[

Since the value of 0 is not included in the interval, we can conclude that at 95% confidence interval there is a difference in the average daily intake of daily products for men and women.

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EXERCISE 1A study was carried out to estimate the mean height in centimeters of 15 years old students in Tunku Abdul Rahman Secondary School. Previous studies indicate that the variance of the height of such students is 80cm. A random sample of 100 students was selected. Suppose that the mean sample was cm.

Find a 95% confidence interval for the mean height of all 15 years old students in the school.

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145x

The average zinc concentration recovered from a sample of zinc measurements in 36 different locations is found to be 2.6 grams per millimeter.

Find a 95% confidence interval for the mean zinc concentration in the river. Assume that the population standard deviation is 0.3

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EXERCISE 2

The contents of 7 similar containers of sulfuric acid are 9.8, 10.2 10.4, 9.8,10.0, 10.2 and 9.6 liters.

Find a 99% confidence interval for the mean of all such containers, assuming an approximate normal distribution.

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EXERCISE 3

An experiment was conducted with two types of engines, A and B . Gas mileages in miles per gallon were measured. Fifty experiments were conducted using engine Type A and 75 experiments were done for Engine Type B. The gasoline used and other conditions were held constant. The average gas mileage for Engine A was 42 miles per gallon and that for Engine B was 36 miles per gallon.

Find a 96% confidence interval for the difference between the gas mileages for the two types of engines. Assume that population standard deviations of gas mileages are 6 and 8 for engines A and B 24

EXERCISE 4