Continuous Random Variables and the Normal Distribution

Transcription

1 Chapter 6 Continuous Random Variables and the Normal Distribution Continuous random variables are used to approximate probabilities where there are many possible outcomes or an infinite number of possible outcomes on a given trial. One of the most well-known continuous distributions used to approximate probabilities is the normal distribution. Traditionally, normal distribution probabilities were obtained using a normal distribution table. These tables are 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 has 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. To generate 30 numbers from a normal distribution with a mean of 45 and a standard deviation of 8 and store them in L 2 : Press. Select 6:randNorm( from the PRB menu. Type 45, 8, 30). Press. Type L 2 and press.

2 Generate 200 numbers from a normal distribution with µ = 100 and σ = 15 and store them in L 3. Generate a histogram of the 200 numbers in L 3 and observe that the histogram is beginning to look like a normal distribution. Experiment with generating a larger number of data values. Computing Normal Distribution Probabilities The computation commands for normal distributions are normalpdf(, normalcdf(, and invnorm(. All three commands can be found by pressing ( ) and selecting the DISTR menu. Computing 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 distribution. There are three different scenarios: Finding the probability that a number will fall between two values under a normal distribution. Finding the probability that a number will fall to the left of a value under a normal distribution. Finding the probability that a number will fall to the right of a value under a normal distribution. The syntax for the normalcdf( function is normalcdf(a, b, µ, σ), where a 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. Finding the Area Between Two Values To find the area between two numbers a and b under the standard normal curve, use the normalcdf( function. Find the probability of getting a value between 1.04 and 1.82 under the standard normal curve. P(1.04 < z < 1.82) = Select normalcdf( from the DISTR menu. Enter 1.04 at the lower: prompt. Enter 1.82 at the upper: prompt. 2

3 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) = 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 the left of b under a normal curve, use the normalcdf( function. The lower bound should be negative infinity (- ). 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 negative number that lies far to the left on the number line. The letter E stands for scientific notation and it is located above the comma key (press ). Find the probability of getting a value less than 0 under the Standard Normal curve. P(z < 0) = 0.5. Select normalcdf( from the DISTR menu. Enter -1E99 at the lower: prompt. Enter 0 at the upper: prompt. Find the probability of getting a value less than under the normal curve with mean 25 and standard deviation 6. The probability is Finding the Area to the Right of a Value To find the area to right of a under the normal curve, use the normalcdf( function. The upper bound should be positive infinity (+ ). Since the TI-84 calculator does not have a built-in key for negative infinity (+ ), use the value 1E99, which represents a positive number that lies far to the right on the number line. The letter E stands for scientific notation and it is located above the comma key (press ). 3

4 Find the probability of getting a value greater than under the standard normal curve. P(z > -1.08) = Select normalcdf( from the DISTR menu. Enter -1E99 at the lower: prompt. Enter 0 at the upper: prompt. Find the probability of getting a value greater than 15.3 under the Normal curve with mean 12 and standard deviation 4. P(z > 15.3) = Inverse Normal Distribution Probabilities There are times in statistics when we have a probability from a normal distribution and need a relevant z-score or raw score. Such computations are called inverse normal probability calculations and can be performed using tables of normal probabilities, but the work is tedious. The TI-84 Plus calculator has a function, invnorm(, that performs these calculations. Suppose we are looking for a number k from a standard normal distribution such that P(z < k) = In other words, what number cuts off an area of to its left in a standard normal distribution? P(z < ) = Select invnorm( from the DISTR menu. Enter at the area: prompt. The invnorm( function always performs calculations based on an area to the left of a number. If you are given an area to the right of a number, find the complementary area on the left by subtracting from 1. 4

5 Find the number x with an area of 0.65 to its right from a normal distribution with a mean of 10 and a standard deviation of 4. That is, find a value x from a normal distribution with the given mean and standard deviation such that P(x < k) = P(x < ) = 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. Instead, the output is the height of the normal distribution above the horizontal axis (the y- coordinate of the graph). 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). Press. (Do NOT forget this step!) To view this graph in the proper window, press Select 1: normalpdf( from the DISTR menu. Press at the area: prompt.. and select 9:ZoomStat. You can trace the curve by pressing. This command may be used to draw any normal distribution curve with any mean and standard deviation. 5

6 Shade the Normal Probability Density Function When calculating the probability of an area under a 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 Press to view the DRAW menu. Select 1: ShadeNorm( from the DRAW menu. Enter 1.04 at the lower: prompt. Enter 1.82 at the upper: prompt. Highlight Draw and press. Example Suppose you wish to shade the area under the standard normal curve that corresponds to 1.04 < z < Begin by turning off all other functions ( ) and statistical plots ( ). Press and adjust the settings to view the standard normal curve, as shown to the left. Find the probability of getting a value greater than 15.3 under the Normal curve with mean 12 and standard deviation 4. Press and adjust the settings as shown to the right. P(x > 15.3) = Notice that the area of the shaded region is also shown on the graph and it is the same value previously calculated from the normalcdf( command. Thus, the ShadeNorm( is an alternative command for normalcdf(, with the added benefit of the shading the area of interest. Optimal window settings are Xmin = μ 3σ, Xmax = μ + 3σ, Xscl = σ, Ymin = -0.2/σ, and Ymax = 0.5/σ. Choose any value you wish for Yscl. 6