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Chapter 6

# Chapter 6

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University of Toronto Scarborough

Statistics

STAB22H3

Moras

Winter

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Chapter 6 Statistical inference draws conclusions about a population or process based on sample data Two types of statistical Inferences: - Confidence interval - Tests of significance This chapter will only consider scenarios of inference about the mean with the standard deviation given. In this setting we can ask questions like: - What is the average loan debt among undergraduate borrowers? - What is the average mile per gallon (mpg) for a hybrid car? - Is the average cholesterol level of undergraduate women at your university below the national average? Overview of inference Purpose of statistical inference is to draw conclusion from data. Probability allows us to take chance variation into account Ex 6.2 Effectiveness of a new drug 20 were given the drug 60% had improvement 20 were given the placebo 40% had improvement Due to probability calculations a difference this large or larger between the results in the two group would occur one time in five simply because of chance variation. Confidence interval used for estimating the value of a population parameter Tests of significance assess the evidence for a claim Both types of inferences are based on sampling distributions of statistics Probability models are most secure, and inference is most reliable when the data are produced by a properly randomized design. Unrealistic assumption: is when the standard deviation () is given. 6.1 Estimating with Confidence Sample mean (x-bar) can be used for mean () of population since it is unbiased But each sample mean (x-bar) can differ from sample to sample An estimate without an indication of its variability is of little value. Statistical confidence n = 500 = 100 www.notesolution.comx = 100 square root 500 = 4.5 Consider: - The 68-95-99.7 rule says that the probability is about 0.95 that x-bar will be within 0 points (2 of the population mean score . - To say that x-bar lies within 9 points of is the same as saying that is within 9 points of x-bar - So 95% of all samples will capture the true in the interval from x-bar 9 to x-bar + 9 The language of statistical inference uses this fact about what would happen in the long run to express our confidence in the results of any one sample. X-bar = 461 95% confidence lies between: 452 and 470 X-bar 9 = 461 9 = 452 and X-bar + 9 = 461 + 9 = 470 2 possibilities of our SRS: 1. The interval between 452 and 470 contains the true . 2. The interval between 452 and 470 does not contain the true . 95% confidence means that the method used gives correct result 95% of the time. Confidence Intervals The interval of # between the values X-bar+9 is called the 95% confidence interval for . X-bar+9 = Estimate + margin of error Margin of error reflects how accurate we believe our guess is based on the variability of the estimate, and how confident we are that the procedure will catch the true population mean . 2 important things about confident intervals 1. It is an interval of the form (a,b), where a and b are numbers computed from the data. 2. It has a property called a confidence level that gives the probability of producing an interval that contains the unknown parameter. Occasionally, 90% and 99% is used, but the 95% rule is mostly used. C stands for confidence level A confidence of 95% C= 0.95 Confidence Interval A level C confidence interval for a parameter is an interval calculated from a sample data be a method that has a probability C of producing an interval containing the true value of the parameter. www.notesolution.com

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