NORMAL DISTRIBUTIONS 1 18 GENERAL INFORMATION ABOUT A DENSITY CURVE 16 14 12 10 8 6 4 2 0 16 14 12 10 8 6 4 2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 a density curve is a smooth curve that models the general shape of a distribution ignoring irregularities or outliers the curve shows what the distribution looks like after very many observations are taken thus it is a model for the population Greek letters are generally used to represent population parameters associated with density curves for example pronounced myoo is used for mean and pronounced sigma is used for standard deviation the horizontal axis is always the data axis just like in the histogram the curve always runs on or above the horizontal axis any curve that is used to model data the total area under any density curve is always 1 sq unit must satisfy these two properties area under a density curve represents percentage of data area under the curve above an interval of values proportion of data in that interval area under a density curve is preserved when a distribution is standardized this happens because z scores retain the position properties of the original data values so percentage of data area under the curve is also retained a density curve is used to find the proportion of data within a certain interval of values as opposed to the proportion of data that are equal to a single value the proportion of data equal to a single value is considered to be zero in a symmetric density curve the mean will be equal to the median in general but not always in a left skewed density curve the mean will be less than the median in general but not always in a right skewed density curve the mean will be greater than the median 2010 Radha Bose Florida State University Department of Statistics NORMAL DISTRIBUTIONS 2 18 1 The pictures below show different symmetric distributions Match the verbal descriptions to the pictures by filling in the table below Verbal Description Picture A B C D I II III IV Verbal Descriptions I The percentage of students who were taller than 5ft 4in given that the average height is about 5ft 5in II The percentage of cars that weighed more than 3500 lbs given that the average weight is about 2800 lbs III The percentage of cases that were processed in less than 3 weeks given that the average processing time is about 2 weeks IV The percentage of the time days out of the year say that the temperature was below 10oC given that the average temperature is about 20oC 2010 Radha Bose Florida State University Department of Statistics NORMAL DISTRIBUTIONS 3 18 Try this on your own matching areas with percentages of data Four pictures of the same symmetric bell shaped distribution with mean 241 are shown below The pictures are only rough sketches they are not drawn to scale Match the following percentages of data to the shaded areas by filling in the table below 10 40 60 90 Picture Percentage of data shaded A B C D A B D C 2010 Radha Bose Florida State University Department of Statistics NORMAL DISTRIBUTIONS 4 18 BELL SHAPED DISTRIBUTIONS values that are within 2 standard deviations of the mean values that have z scores between 2 and 2 are considered to be typical or not unusual values that are more than 3 standard deviations away from the mean values that have z scores less than 3 or greater than 3 are considered to be very far away from the mean or unusual they are far from the standard or far from the norm Empirical Rule approx 68 of all observations fall within 1 standard deviation of the mean 95 of all observations fall within 2 standard deviations of the mean 99 7 of all observations fall within 3 standard deviations of the mean Since percentage of data is represented by area under the curve we can use the above percentages to divide up the area under a bell shaped curve as shown below We use the symmetry of the curve to help us with the break up of percentages Values 3 2 2 3 Z scores 3 2 1 0 1 2 3 unusual values typical values unusual values 2010 Radha Bose Florida State University Department of Statistics NORMAL DISTRIBUTIONS 5 18 2 The figures below though not entirely accurate were inspired by data obtained at http ndb nal usda gov ndb beef show and will suffice for the purposes of this exercise Let us suppose that the amounts of cholesterol in one pound samples of 80 lean ground beef are Normally distributed with mean 322mg and standard deviation 14mg Sub pound samples Obs Dis a Half of all pound samples are predicted to have more than mg of cholesterol b What percent of pound samples can we expect to have 321 mg of cholesterol c Draw an appropriately labeled density curve showing the break up of the data according to the Empirical rule then answer the questions that follow d What percent of pound samples are expected to have less than 280 mg of cholesterol e What percent of pound samples are expected to have between 294 and 322 mg of cholesterol f The middle 95 of cholesterol amounts are predicted to be between mg and mg g We predict that the highest 2 5 of cholesterol amounts will be above mg h The 84th percentile of a distribution is a number that has 84 of the data less than or equal to itself The 84th percentile of the cholesterol amounts is expected to be mg i One sample had 400 mg of cholesterol Was this a typical or an unusual amount j One sample had 300 mg of cholesterol Was this a typical or an unusual amount 2010 Radha Bose Florida State University Department of Statistics NORMAL DISTRIBUTIONS 6 18 Try this on your own Empirical rule Let us suppose that the number of hours per week that pre kindergarten children who go to daycare spend in daycare can be described by a Normal model with mean 30 hours and standard deviation 7 hours Sub Obs Dis a Draw an appropriately labeled density curve showing what the Empirical rule predicts about the number of hours per week spent in daycare b Typically pre k children spend between and hours per week in daycare c What percent of pre kindergarten children are expected to spend between 23 and 44 hours per week in daycare d What percent of pre kindergarten children are predicted to spend more than 16 hours a week in daycare e A warning report is automatically generated if a child ends up spending an unusually high number of hours per week in daycare A report will therefore be generated for children who spend more than hours per week in daycare f The 84th percentile of the times spent in daycare per week is approximately g What is the 16th percentile