Meeting 06 Confidence Intervals - Ijaz.pptx

Meeting 06 Confidence Intervals - Ijaz.pptx - Texas A&M...

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Texas A&M University ENGR 112 – Spring 2018 Foundations of Engineering II Meeting 06 Confidence Intervals 1
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Agenda Review Histogram, Normal Distributions, Standard Normal Distribution Review use of z-table calculations How to use normpdf , normcdf , and norminv in MATLAB Define Sampling Distribution Define confidence interval Central Limit Theorem How to compute a confidence interval on the mean with a specified degree of confidence given the population standard deviation and test data. How to make simple pdf and cdf plots 2
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Learning Objectives Use z-table calculations to compute a confidence interval on the mean with a specified degree of confidence given the population standard deviation and test data. How to use normpdf , normcdf , and norminv in MATLAB How to make simple pdf and cdf plots 3
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A few terms used in Statistics Trial Getting a data point (or a set of data points) Event Outcome of a trial Probability Measure of the likelihood that an event will occur. 1 Bin is also called Class. Bin size is also called Class Interval 1 (Reference: Wikipedia; visited 02/08/2017)
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Consider the peanut crop data Image source: - 2009-02-20 Characterizing batches of peanuts coming from farms
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Peanut Crop Major Diameter Histogram 0 10 20 30 40 50 60 70 Frequency
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0 10 20 30 40 50 60 70 Frequency Probability Histograms can be used to determine the probability of a value falling into a given classification. Given the histogram illustrating the sample of 231 peanuts, what is the probability of sampling a peanut with a major diameter in the range of 19.51- 20.89 mm? range frequency 11.23 - 12.61 1 12.61 - 13.99 2 13.99 - 15.37 12 15.37 - 16.75 43 16.75 - 18.13 57 18.13 - 19.51 65 19.51 - 20.89 31 20.89 - 22.27 15 22.27 - 23.65 4 23.65 - 25.03 1
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Probability Sometimes Histograms are drawn using Relative Frequencies (or Probabilities) instead of Frequencies; where Relative Frequency is obtained by dividing the Frequency by the number of data points in the sample. The probability of sampling a peanut with a major diameter in the range of 19.51- 20.89 mm can be readily determined. range relative frequency 11.23 - 12.61 0.004329004 12.61 - 13.99 0.008658009 13.99 - 15.37 0.051948052 15.37 - 16.75 0.186147186 16.75 - 18.13 0.246753247 18.13 - 19.51 0.281385281 19.51 - 20.89 0.134199134 20.89 - 22.27 0.064935065 22.27 - 23.65 0.017316017 23.65 - 25.03 0.004329004
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Probability Let’s define Probability Density to be ‘Relative Frequency divided by the size of the bin’. Now if Histogram is drawn using Probability Density, the probability for a trial to have a value within a given bin is the area of the bar (or rectangle) drawn at that bin. The probability of sampling a peanut with a major diameter in the range of 19.51- 20.89 mm can be determined as follows. range density 11.23 - 12.61 0.003137 12.61 - 13.99 0.006274 13.99 - 15.37 0.037644 15.37 - 16.75 0.134889 16.75 - 18.13 0.178807 18.13 - 19.51 0.203902 19.51 - 20.89 0.097246 20.89 - 22.27 0.047054 22.27 - 23.65 0.012548 23.65 - 25.03 0.003137
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Probability Density Function As we decrease the bin size (and hence increase the number of bins) the Probability Density histogram approaches a continuous function, say f(x).
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Continuous Distributions Histograms (a frequency distribution) can be represented by means of a smooth curve commonly called a continuous distribution (probability density function, PDF).
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