Z-tables. January 12, This tutorial covers how to find areas under normal distributions using a z-table.

Transcription

1 Z-tables January 12, 2019 Contents The standard normal distribution Areas above Areas below the mean Areas between two values of Finding -scores from areas Z tables in R: Questions This tutorial covers how to find areas under normal distributions using a -table The standard normal distribution Thanks to the central limit theorem distributions of means often fall into a normal bellshaped distribution Since we ll be dealing with means as dependent measures a lot this quarter and in our research, we ll need to be familiar with the properties of the normal distribution All normal distributions have the same shape They only differ by their means and standar deviations The general equation for the normal probability distribution is: e (x µ)2 2πσ Where µ is the mean and σ is the standard deviation of the distribution (It s kind of remarkable that this ubiquitous function has two famous transcendental numbers in it, e, and π, plus the irrational number 2) We choose one particular normal distribution, the standard normal, as a reference for tables The standard normal distribution, or -distribution has a mean of ero and a standard deviation of 1 The standard normal s probability distribution function simplifies to: e x2 2π It looks like this: 1

2 Here are some exercises on using the -table to find areas under this standard normal distribution (either in the book, Excel spreadsheet, handout, R, or one of many websites or statistics programs) We ll start with an easy one: What is the area under the standard normal distribution above =0? The area is shaded in the figure below: area =05 0 The answer is 05 because the normal distribution has a total area of 1 and is symmetric about the mean of 0 Areas above Example: find the area above =1 The area is the shaded region below: 2

3 area =01587 The area can be found by using table A in the book Find the value in the first column for =1 The third column gives the area under the standard normal above The relevant part of the table should look something like this: 1 Area between mean and Area beyond On the row where the first column as = 1, the third column shows that the area under the curve above is The middle column is the area between ero and Since right-half of the area is 05, you can see that columns 2 and 3 add up to 05 (for =1, = 05) Areas below the mean Example: What is the area under the standard normal distribution below = -2? 3

4 area = Notice that the -table doesn t show areas for negative values of That s because the -distribution is symmetrical, so for our example, the area below = -2 is the same as the area above = 2: area =00228 The area above = 2 can be found in the table: 2 Area between mean and Area beyond Example: Find the area under the standard normal distribution below = 1: 4

5 area =08413 There are a couple of ways to do this one One way is to realie that since the total area is 1, the area below = 1 is equal to 1 minus the area above = 1 which we know from before is So the area below 1 is = Another way to do this is to see that the area below 1 is the sum of the area between ero and 1 and the area below ero which is 05 From second column in the table, the area between ero and 1 is So the total area is = Areas between two values of Example: What is the area under the standard normal distribution between 1 and 2? area = The trick is to understand that the area can be computed by subtracting the area above = 2 from the area above = 1: 5

6 area = area =00228 The difference is = Example: What is the area under the standard normal between = -2 and 1? 2 area = Again, there are a couple of ways to solve this one One way is to use the fact that the total 6

7 area is 1, so the area between -2 and 1 is equal to 1 minus the areas in the tails The area below = -2 is and the area above 1 is 01587: area = area =01587 So the total area is equal to = Another way to solve this one is to use the second column in table, which is the area between the mean and The area between = -2 and = 0 is the same as the area between = 0 and = 2, which according to the table is The table also tells us that the area between 0 and 1 is 03413: 1 7

8 area = area = So the total area is = Finding -scores from areas Example: Find the score for which 5% of the area under the standard normal distribution lies above 8

9 area = To solve this one we need to find the row in the table for which the third column, the area beyond, is nearest to 005: Area between mean and Area beyond So the answer is = 164 Example: Find the value of for which 10% of the area under the standard normal distribution lies below: area = We ll use the fact that the normal distribution is symmetrical, and find the -value for which 10% lies above: 9

10 area = Area between mean and Area beyond The closest value of is 128 Using symmetry, we know that 10% of the area under the standard normal distribution lies below = -128 Example: Find the values of that bracket the middle 95% of the area under the standard normal distribution area = The middle 95% of the area leaves (100-95)/2 = 25% in each of the two tails; 10

11 area =0025 area = So we need to find the -value for which the area above is 0025 Area between mean and Area beyond From the table, = 196 Therefore 95% of the area under the standard normal distribution lies between = -196 and =

12 Z tables in R: R has two functions that give you values in the -table pnorm converts -scores to areas, and qnorm converts areas to -scores The R commands shown below can be found here: TableR # The -distribution # # R s function pnorm calculates the area under the normal distribution # below a -score By default, it uses a standard normal distribution # (mean 0, sd 1) # # For example, to find the area under the standard normal below =1, we use: pnorm(1) [1] # Note that the tables in the book and the Excel spreadsheet give you # the areas ABOVE To get the values in these tables we need to subtract # the result from pnorm from 1: 1-pnorm(1) [1] # You should see that this value matches the value in the third column in # the standard normal table for =1 # The area under the standard normal below = -2 is: pnorm(-2) [1] # The area under the standard normal between =1 and =2 can be found # by finding the area below =2 and subtracting the area below =1: pnorm(2) - pnorm(1) [1] # The area between = -2 and =1 can be caculated similarly: pnorm(1) - pnorm(-2) [1] # To go the other way and find -scores from areas we use # the function qnorm # # For example, the -score for which 95% of the area of the # standard normal falls below is: qnorm(95) [1] # This is, of course the same as the -score for which 5% falls above 12

13 # To find the -score for which 10% of the area lies below we use: qnorm(1) [1] # To find the values of that bracket the middle 95% is found # by first finding the lower -score This is the -score # for which only 25% falls below: qnorm(025) [1] # Since the -distribution is symmetric, the upper value of # is just the positive sign of this: -qnorm(025) [1] # In general, if you want to find the scores that bracket # the middle p% we can first set a variable p to some value: p <- 5 # And then use c to concatenate the two -scores into a single # vector: c(qnorm((1-p)/2),-qnorm((1-p)/2)) [1] Questions Now it s your turn Here are 30 random -distribution problems and answer (including R commands) Draw pictures if it helps 1) Find the area under the standard normal distribution below = 120 pnorm(12) [1] Answer: ) Find the value of for which 93 percent of the area under standard normal distribution lies above qnorm(1-093) [1] Answer: = ) Find the area under the standard normal distribution between = 010 and =

14 pnorm(06) - pnorm(01) [1] Answer: ) Find the value of for which 5 percent of the area under standard normal distribution lies below qnorm(005) [1] Answer: = ) Find the value of for which 1 percent of the area under standard normal distribution lies above qnorm(1-001) [1] Answer: = 233 6) Find the area under the standard normal distribution below = 040 pnorm(04) [1] Answer: ) Find the area under the standard normal distribution above = pnorm(-07) [1] Answer: ) Find the value of for which 85 percent of the area under standard normal distribution lies above qnorm(1-085) [1] Answer: = ) Find the value of for which 65 percent of the area under standard normal distribution lies below qnorm(065) [1]

15 Answer: = ) Find the area under the standard normal distribution above = pnorm(17) [1] Answer: ) Find the value of for which 26 percent of the area under standard normal distribution lies above qnorm(1-026) [1] Answer: = ) Find the area under the standard normal distribution above = pnorm(01) [1] Answer: ) Find the area under the standard normal distribution between = 030 and = 180 pnorm(18) - pnorm(03) [1] Answer: ) Find the value of for which 37 percent of the area under standard normal distribution lies above qnorm(1-037) [1] Answer: = ) Find the area under the standard normal distribution above = pnorm(09) [1] Answer:

16 16) Find the value of for which 65 percent of the area under standard normal distribution lies above qnorm(1-065) [1] Answer: = ) Find the area under the standard normal distribution between = -140 and = -070 pnorm(-07) - pnorm(-14) [1] Answer: ) Find the value of for which 44 percent of the area under standard normal distribution lies below qnorm(044) [1] Answer: = ) Find the range of values which covers the middle 24 percent of the area under the standard normal distribution c(qnorm(1-(1-024)/2),-qnorm(1-(1-024)/2)) [1] Answer: Between = -031 and = ) Find the area under the standard normal distribution between = -060 and = 120 pnorm(12) - pnorm(-06) [1] Answer: ) Find the area under the standard normal distribution between = -040 and = 060 pnorm(06) - pnorm(-04) [1] Answer: ) Find the value of for which 6 percent of the area under standard normal distribution lies above 16

17 qnorm(1-006) [1] Answer: = ) Find the range of values which covers the middle 52 percent of the area under the standard normal distribution c(qnorm(1-(1-052)/2),-qnorm(1-(1-052)/2)) [1] Answer: Between = -071 and = ) Find the area under the standard normal distribution above = pnorm(-01) [1] Answer: ) Find the area under the standard normal distribution below = -130 pnorm(-13) [1] Answer: ) Find the area under the standard normal distribution above = pnorm(-13) [1] Answer: ) Find the area under the standard normal distribution between = -000 and = 100 pnorm(1) - pnorm(-0) [1] Answer: ) Find the area under the standard normal distribution below = -080 pnorm(-08) [1] Answer:

18 29) Find the range of values which covers the middle 46 percent of the area under the standard normal distribution c(qnorm(1-(1-046)/2),-qnorm(1-(1-046)/2)) [1] Answer: Between = -061 and = ) Find the value of for which 76 percent of the area under standard normal distribution lies below qnorm(076) [1] Answer: =