The Normal Distribution

Transcription

1 Chapter 6 The Normal Distribution Continuous random variables are used to approximate probabilities where there are many possibilities or an infinite number of possibilities on a given trial. One of the most well-known continuous distributions used to approximate probabilities is the normal Traditionally normal distribution probabilities were figured using a normal distribution table. The table method is being replaced with calculators such as the TI-84 Plus. The calculator reduces the time needed to perform the calculations and reduces the rounding errors that occur because of the brevity of the tables in elementary statistics textbooks. Normal Distribution Randomly Generating a Number From a Normal Distribution Just as the TI-84 had a built-in function to generate random real numbers from a Binomial distribution, it also has a built-in function to generate random real numbers from a specific Normal distribution with a mean µ and standard deviation σ. The random real numbers represent x values. The general syntax is randnorm(µ, σ, n), where n is the number of random real numbers. The following command will generate 30 numbers from a Normal distribution with a mean of 45 and a standard deviation of 8 and store them in L2. Select MATH > PRB > 6:randNorm( and press ENTER. Type: 45, 8, 30) > STO > L2 Generate 200 numbers from a Normal distribution with µ = 100 and σ = 15 and store them in L3.

2 2 Graphing Calculator Manual Generate a histogram of the 200 numbers in L3 and observe that the histogram is beginning to look like a normal Experiment with generating a larger number of data values. Computing Normal Distribution Probabilities The commands for the Normal distribution are normalpdf(, normalcdf(, and invnorm(. They are located on the DISTR page. DISTR appears above the VARS key. Compute Cumulative Normal Probabilities The normalcdf( function stands for normal cumulative density function and gives the probability of getting an x value that falls within an interval of values from the normal There are three possibilities: Finding the probability that a number will fall between two values under the Normal Finding the probability that a number will fall to the left of a value under the Normal Finding the probability that a number will fall to the right of a value under the Normal The syntax for the normalcdf( function is normalcdf(l, B, µ, σ), where L is the lower bound of the interval, B is the upper bound of the interval, µ is the mean, and σ is the standard deviation. The values for µ and σ may be omitted if it is the Standard Normal Finding the Area Between Two Values To find the area between two numbers a and b under the Standard Normal curve, P(a < z < b) = normalcdf(a, b, 0, 1). Find the probability of getting a value between 1.04 and 1.82 under the Standard Normal curve. Type: 1.04, 1.82, 0, 1) and press ENTER. P(1.04 < z < 1.82) = Find the probability of getting a value between 0 and 3 under the Standard Normal curve. Find the probability of getting a value between 10 and 13 under the Normal curve with a mean of 10 and a standard deviation of 2. P(10 < x < 13) = 0.43

3 Chapter 6: The Normal Distribution 3 Find the probability of getting a value between 2 and 12 under the Normal curve with a mean of 10 and a standard deviation of 2. Finding the Area to the Left of a Value To find the area to left of b under the Normal curve, P(z < b) = normalcdf(-, b, µ, σ). The problem is that the TI-84 calculator does not have a built-in key for negative infinity (- ). Thus, the value -1E99 is used, which represents a very large negative number. The letter E stands for scientific notation and it is located above the comma (,) key (2 nd >,). Thus, the command will look like: normalcdf(-1e99, b, µ, σ). Find the probability of getting a value less than 0 under the Standard Normal curve. Type: -1 > 2 nd >, > 99,0) and press ENTER. P(z < 0) = 0.5. Find the probability of getting a value less than under the Normal curve with mean 25 and standard deviation 6. Finding the Area to the Right of a Value To find the area to right of a under the Normal curve, P(z > a) = normalcdf(a, 1E99, µ, σ). Find the probability of getting a value greater than under the Standard Normal curve. Type: -1 > 2 nd >, > 99,0) and press ENTER. P(z > -1.08) = Find the probability of getting a value greater than 15.3 under the Normal curve with mean 12 and standard deviation 4. Inverse Normal Distribution Probabilities There are times in statistics when we have a probability and need a relevant z-score or raw score. The problem of this type may look like: P(z >?) = Such problems are known as inverse normal distribution problems. Such computations can be performed using tables of normal probabilities, but the work is tedious, error-prone, and often has rounding errors.

4 4 Graphing Calculator Manual Fortunately, the calculator has a function, invnorm(, that performs the calculation. We know from the previous section that the unknown in P(z >?) = is Select: 2 nd > VARS > 3:invNorm( and press ENTER. Type: ) and press ENTER. The screen is telling us that the answer is positive The invnorm( function gives an answer based on a cumulative probability of from - to Since the Normal distribution is symmetric, the same cumulative probability applies to to. It is always advisable to draw the normal curve to help in visualizing this concept. Graph the Normal Probability Density Function The function normalpdf( stands for Normal probability density function and does not actually generate a probability, since it applies to a single x value in a continuous distribution and that probability is always zero. The main use of this command is to draw the Normal curve. The syntax for the function is normalpdf(x, μ, σ), where μ is the mean and σ is the standard deviation. The following sequence of commands will draw the standard normal curve (μ =0 and σ = 1). Select: Y = > 2 nd > VARS > 1: normalpdf( and press ENTER. Type: x, 0, 1) > ZOOM > 9 This command may be used to draw any Normal distribution curve with any mean and standard deviation. Shade the Normal Probability Density Function When calculating the probability of an area under the Normal curve, it is often helpful to shade the area. The syntax for the TI-84 Plus command to do this is ShadeNorm(a, b, µ, σ). Example

5 Chapter 6: The Normal Distribution 5 To shade the area under the Standard Normal curve for P(1.04 < z < 1.82) = 0.115, begin by turning off all other graphs (STATPLOT or Y =). Adjust the WINDOW to view the Standard Normal curve, as shown on the right. Select: 2 nd > VARS > DRAW > 1: ShadeNorm( and press ENTER. Type: 1.04, 1.82, 0, 1) and press ENTER. Notice that the area of the shaded region is also shown on the graph and it is the same value calculated from the normalcdf( command. Thus, the ShadeNorm( is an alternative command for normalcdf(, with the added benefit of the shading of the area. Example Find the probability of getting a value greater than 15.3 under the Normal curve with mean 12 and standard deviation 4. Adjust the WINDOW as shown on the right. Type: ShadeNorm(15.3, 1E99, 12, 4) and press ENTER. P(x > 15.3) =