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To the world you may be one person, but to one person you may be the world. Probability The probability of an event is the proportion of times we would expect the event to occur in an infinitely long series of identical sampling experiments. Probability If all the possible outcomes are equally likely, the probability of the occurrence of an event is equal to the proportion of the possible outcomes characterized by the event. Probability Probability is a very useful notion in situations involving at least some degree of uncertainty; it gives us a way of expressing the degree of assurance that a particular event will occur. Probability Chance factors inherent in forming samples always affect sample results. Sample results must therefore be interpreted with that in mind. Probability The probability of occurrence of either one event OR another OR… is obtained by adding their individual probabilities, provided the events are mutually exclusive. Probability The probability of the joint occurrence of one event AND another AND another AND another… is obtained by multiplying their separate probabilities, provided the events are independent. Probability Any relative frequency distribution may be interpreted as a probability distribution. Probability Grade Relative Frequency A .15 B .30 C .40 D .10 F .05 Pr = .15 + .30 + .40 = .85 (85% chance) Confidence Statement Statistic ± Margin of Error Margin of Error Confidence Interval A range of values constructed from sample data so the parameter occurs within that range at a specified probability. The specified level of probability is called the level of confidence. Confidence Interval for a Sample Mean Confidence Interval 95% Confidence Interval 95% of those surveyed will fall into a certain range surrounding the mean Confidence Interval The average size of a mortgage applied for in 1993 was $116,991 as opposed to $119,999 in 1992. A sample of 64 mortgages showed that the standard deviation of the amount applied for was $6019. Find a 95% confidence interval for the average size of a mortgage applied for in 1993. Confidence Interval n = 64 x = $116,991 s = 6019 Ci = 95% Z = 1.96 Confidence Interval X ± Z( s / √n) = 116,991 ± 1.96( 6019 / √64 ) = 116,991 ± 1.96(752.38) = 116,991 ± 1474.66 = 115516.34 to 118465.66 Confidence Interval According to the Family Economic Research Group of the US Department of Agriculture, middle income couples who had babies in 1992 will spend an average of $128,670 by the time the baby is 18 years old. Assume the standard deviation of a sample of 100 families was $8473. Find a 90% confidence interval for the average cost to raise a child born in 1992. Confidence Interval n = 100 s = 8473 X = 128,670 Ci = 90% Z = 1.645 Confidence Interval X ± Z( s / √n) = 128,670 ± 1.645( 8473 / √100) = 128,670 ± 1.645(847.3) = 128670 ± 1393.81 = 127276.19 to 130063.81 Confidence Interval for a Sample Proportion p p(1-p) n Confidence Interval for a Sample Proportion Suppose 1,600 of 2000 union members sampled said they plan to vote for the proposal to merge with the UMA. Using the .95 level of confidence, what is the interval estimate for the population proportion? Based on the confidence interval,what conclusion can be drawn? Confidence Interval for a Sample Proportion p+z P(1-p) n = .80 + 1.96 =.80 + 1.96 .80(1-.80) 2,000 .00008 = .782 and .818 Point Estimate A value, computed from sample information that is used to estimate the population parameter