The document provides information about statistics and economics tutorials being offered after school, including regression analysis, correlation, and the normal distribution. It gives examples of calculating rank correlation, finding regression equations, and using the standard normal distribution table. It also explains key aspects of the normal distribution like the 68-95-99.7 rule and how to calculate probabilities using the normal distribution function in Excel.

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3. Find rank correlation between the following?

5. Solution … 36 1 0 1 9 1 16 4 4 diff. sq 1 0 1 -3 1 -4 2 2 difference 7 1 2 8 3 6 4 5 rank of y 8 1 3 5 4 2 6 7 rank of x

8. solution 0.405473 562485 557095 29501.2 2.29 8.04 3.26 0 0 172.1 171.4 296207 293860 295012 22.9 80.4 32.6 1721 1714 29929 28900 29410 0.81 1.96 -1.26 0.9 -1.4 173 170 28900 28561 28730 4.41 5.76 5.04 -2.1 -2.4 170 169 30625 30976 30800 8.41 21.16 13.34 2.9 4.6 175 176 29929 30276 30102 0.81 6.76 2.34 0.9 2.6 173 174 28900 29241 29070 4.41 0.16 0.84 -2.1 -0.4 170 171 29929 29584 29756 0.81 0.36 0.54 0.9 0.6 173 172 29241 28224 28728 1.21 11.56 3.74 -1.1 -3.4 171 168 29241 27889 28557 1.21 19.36 4.84 -1.1 -4.4 171 167 29584 29584 29584 0.01 0.36 -0.06 -0.1 0.6 172 172 29929 30625 30275 0.81 12.96 3.24 0.9 3.6 173 175 Y^2 X ^2 XY dy^2 dx^2 dxdy dy dx y X

18. NORMAL DISTRIBUTION … 1/10 2/10 4/10 2/10 1/10 Values of a variable, say test scores 60 70 80 90 In this example 10 people took a test. The height of each bar is the relative frequency or percentage of those in that range of scores. What % of people had test scores between 70 and 80? 40% What % of people had scores less than 70? 30% If you add up all the fractions what do you get? 1

22. circles and density A a 25% of the area is in A on the large circle and 25% of the area of the small circle is in part a. How can they both be 25%? It is 25 % of its own total. There are as many different normal distributions as there are circles. BUT, normal distributions are divided up, not into quarters, but in another way.

25. On the previous screen we see a graph of a normal distribution. Let’s consider an example to highlight some points. Say a company has developed a new tire for cars. In testing the tire it has been determined that the mean tire mileage is 36,500 miles and the standard deviation is 5000 miles. Along the horizontal axis we measure tire mileage. The normal distribution rises above the axis. Note the highest point of the curve occurs above the mean - in our tire example we would be at 36,500. On the curve we have two inflection points, and these occur 1 standard deviation away from the mean. So, mileages 31,500 and 41,500 are 1 standard deviation for the mean and the inflection points occur above them.

33. Note about normal distribution: 1. There are many normal distributions, each characterized by a mean value and a standard deviation. 2. The high point of the curve is above the mean and for a normal distribution the mean = median = mode. 3. Depending on the variable, the mean can be negative, zero, or positive. 4. The normal curve is symmetric. This means each side is a mirror image of itself. 5. Larger standard deviations result in a flatter, wider distribution. 6. Probabilities for the variable are found from areas under the curve - the 65, 95, 99.7 rule is an example of this.

34. miles 26,500 31,500 36,500 41,500 46,500 -2 -1 0 1 2 z Remember the concept of a z score from earlier. z = ( a value minus the mean)/standard deviation. So the value 26,500 has a z = (26,500 - 36,500)/5000 = -2. This means 26,500 is 2 standard deviations below the mean. You can check the other values.

35. The standard normal distribution Remember how we said there are many different circles and many different normal distribution? Sure you do. The z value translates any normally distributed variable into what is called the standard normal variable. Technically the picture I have on the previous screen is misleading because the z’s are a different scale than the miles, but don’t worry. In the book there is a table with z values and areas under the curve. Let’s see how to use the table. Here is one place where I want you to be extra careful when you calculate z. Round z to 2 decimal places. The z value is broken up into two parts a.b and .0c. when added we get a.bc. For example the number 2.13 is broken up into 2.1 and .03

36. Using the standard normal table The z = 2.13 means we should go down the table to 2.1 and then over to .03. The number in the table is .9834. This means the probability of getting a value less than z = 2.13 is 98.34%. In the tire example if we look at the mean value 36,500, we see the z = (36,500 - 36,500)/5000 = 0.00 and in the table we see the value .5000. Thus, there is a 50% chance the tire mileage will be less than 36,500. So the table has the area under the curve to the left of the value of interest. We may want other z’s and other areas. What do we do?

37. Say we want the area to the right of a z that is greater than 0? The table has the area to the left. Whatever the z is, go into the table and get the area and then take 1 minus the area in the table. a b The z here would be negative. Say we want area b. Area a is in the table and b is 1 minus area a. Area b would be found in a similar way to what is above.

38. Back in the old days when I had to walk to school uphill both ways in three feet of snow, the standard normal table was all we had to calculate probabilities for a normal distribution. Now we have Microsoft Excel to make the calculations. The NORMSDIST function assumes we have a z value and we want to find the area the the left of the z - the area to the left is the cumulative probability. The function has the form =NORMSDIST(z), where z is the value we have. z can be negative in Excel. The NORMDIST function allows us to just work with the variable without getting the z and we can still have the cumulative probability. The function has the form =NORMDIST(value, mean, standard deviation, TRUE). This is an innovation of Excel over the old days.

39. Sometimes we may have an area and want to know the z. The function NORMSINV asks us to give an area to the left of a value and the function will give us the z value. The form of the function is =NORMSINV(cumulative probability). The function NORMINV does the same, except not in z value form. It just give the value in the same form as the variable. The form of the function is =NORMINV(cumulative prob, mean, standard deviation)

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