1.1 Populations and Samples

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1.1 Populations and Samples

• Population Parameters and Sample Statistics
  Population
• Simple Random Sampling
• (3) Population Size

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(1) Population Parameters and Sample Statistics
Population
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(2) Simple Random Sampling

• RAN key
• construct a random number table

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Answer (for reference)

• 0.871 0.843 0.874 0.237 0.451 0.770 0.962 0.980
• 0.583 0.201 0.199 0.565 0.298 0.830 0.727 0.690
• 0.532 0.932 0.508 0.710 0.900 0.661 0.481 0.484
• 0.561 0.119 0.206 0.364 0.814 0.366 0.964 0.703

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Population Size

• N is larger than or equal to 10.
• Notice Size of a population need not be
  large.

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1.2 Central Tendency

1. mean
2. median
3. mode

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1. mean

• Population Mean for Ungrouped Data
• Sample Mean for Ungrouped Data
• Population Mean for Grouped Data
• Sample Mean for Grouped Data
• Combined Sample Mean
• The Weighted Mean

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2. median

• Median of data arranged in order of magnitude
• The median
• Median Found by Ungrouped Frequency Distribution
• Median Found by Grouped Frequency Distribution

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(i) Median Found by Ungrouped Frequency
Distribution
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(ii) Median Found by Grouped Frequency
Distribution
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3.mode

• In general, the mode of a set of data is the
  value which occurs most frequently in the set.

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1.3 Measures of Dispersion (Dispersion
and Varibility)

1. The Range
2. The Interquartile Range, the Five-number summary
   and box plots
3. The Variance and the Standard Deviation

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1.The Range

• The range of a set of data is the difference
  between the largest value and the smallest value
  of the set.
• Example 1
• Find the ranges of the following sets of data
• A 40, 41, 42, 58,59,60
• B 20,20.1,20.2,59.1,59.2,60
• C 10,20,30,40,50,60,70
• The range of A
• The range of B
• The range of C

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2.The Interquartile Range, the Five-number
summary and box plots

• Lower Quartile Q1 value
• Median Q2 value
• Upper Quartile Q3 value
• The Interquartile Range Upper Quartile Lower
  Quartile
• Q3- Q1

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Example 6

• Find the interquartile range of the following set
  of numbers.
• 2,3,3,9,6,6,12,11,8,2,3,5,7,5,4,4,5,12,9

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Example 7

• The table gives the cumulative distribution of
  the heights (in cm) of 400 children in a certain
  school
• Find
• (i)Draw a cumulative frequency curve.
• (ii)Estimate the median.
• (iii)Determine the interquartile range.

Height(cm) lt100 lt110 lt120 lt130 lt140 lt150 lt160 lt170
Cumulative Frequency 0 27 85 215 320 370 395 400
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The Five-number summary and box plots

• The format of Box and Whiskers diagram is shown
  below

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C.W. Q3

• The weekly expenditure on soft drinks of 20
  football players re given in the following table
• Expenditure on soft drinks of 20 football
  players(in dollars)
• Find
• Maximum value
• Upper quartile
• Median
• Lower quartile
• Minimum value
• and then draw the Box and Whiskers diagram

26 34 31 36 16
27 32 21 43 41
30 6 15 38 27
58 35 28 21 20
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3.The Variance and the Standard Deviation

• (I)Population variance and population standard
  deviation
• (II)Sample Variance and Sample Standard
  Deviation

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(I) Population variance and population
standard deviation

• Population variance, ?2
• Population Standard Deviation, ?

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(II) Sample Variance and Sample Standard
Deviation

• Sample Variance s2

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Example 1

• Two machines, A and B, are used to pack biscuits.
  A sample of 10 packets was taken from each
  machine and the mass of each packet, measured to
  the nearest gram, was noted.
• Machine A(mass in g) 196,198,198,199,200,200,201,
  201,202,205
• Machine B(mass in g)
• 192, 194, 195, 198, 200, 201, 203, 204, 206,
  207
• (i) Find the standard deviation of the masses of
  the packets taken in the sample for each
  machine. (ii)Comment on your answer.

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C.W.mean ,standard deviation

• The following are two sets of data of an
  experiment obtained by two different
  studentsVolume of acid measured (cm3)
• Student A
• 8,12,7,9,3,10,12,11,12,14
• Student B
• 7,6,7,15,12,11,9,9,13,11
• (i) What is the mean volume of acid measured by
  each student?(ii) What is the standard
  deviation?(iii) Which set of results is more
  reliable?