units into z-units by using the equation. of 115, how much smarter is he than everybody else?įirst we translate the I.Q. scores have a normal distribution, approximately, with a mean of 100 and a standard deviation of 16. (Just like calendar days are a distribution where the scale is approximately the time it takes for earth to make one complete orbit around the sun)Īnd we can relate z directly to the normal distribution and so we know the percentile of the score What standard score cuts off 9.7% of the distribution to the left of score? (We know 100%-(2*9.7)%=80.6%, so we find z = 1.30, and we want the negative side so z = -1.30, or 1.3 standard deviations to the left)ĭefinition: z is a distribution that is scaled in standard units where the unit is standard deviations from the mean in a normal distribution. What percent of the standard normal curve lies below 1.75 standard deviations? (For 1.75 z, the area = 91.99%, 91.99/2 + 50 = 95.99%) The table in the book gives the area in percentages bounded on the positive and negative side by the z-value (which is equivalent in units like this: 1 z = 1 S.D.) Remember area under the curve is 100% 50% of the curve lies to either side of the mean The book gives many examples and lots of practice-good idea We know at any point on the x-axis what the density or area under the curve is Important properties of this line and the shape it createsĪrea under the curve is 100%-at 0 height is about 40%-draw two triangles and calculate area (1/2*40*± 3) as an approximation of the curve The result is what is called a bell-shaped curve You know the formula for a straight line: y = bx + c The normal curve is also a line but it is defined by: The normal distribution, also called the z-distribution Example: How frequently do you go to the student store on a 1 to 7 scale? How many days per week is that? Does knowing the standard units of days give us more information? Standard scales convey important information in addition to the value observed. We are now going to learn about a distribution (the z-distribution) that has some very handy attributes and we are going to learn how to translate our observations, scores, distributions, into that scale and back again We say URSA instead of the formal name (University Records System Access) to communicate We translate feet into inches to make addition easier, e.g. We already translate units all the time for convenience. This chapter is about taking advantage of something we know is true to give us an edge in making decisions-we do this by translation Find the 75th percentile (a.k.a., third (upper) quartile) of package weights.2004, S.Find the 25th percentile (a.k.a., first (lower) quartile) of package weights.Find the weight that must be printed on the packages. Suppose that the company only wants 1% of packages to be underweight.Estimate the probability that a package weighs between 47.9 and 53.0 grams.Estimate the probability that a package weighs less than the printed weight of 47.9 grams.Why wouldn’t the company print the mean weight of 49.8 grams as the weight on the package?. It is helpful to draw two axes: one in the measurement units of the variable, and one in standardized units. Sketch the distribution of package weights.Suppose package weights have an approximate Normal distribution with a mean of 49.8 grams and a standard deviation of 1.3 grams. Naturally, the weights of individual packages vary somewhat. The wrapper of a package of candy lists a weight of 47.9 grams. That is, there are inflection points at \(\mu\pm \sigma\).
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