Chapter 6: The Normal Probability Distribution This chapter is to introduce you to the concepts of normal distributions.  E.g. if a large number of students took a statistics exam and their marks were recorded in a frequency distribution table, we might expect a normal distribution. Some students will do very well, some will do very badly, but the majority of the students will fall close to either side of the mean.

Normal probability distribution characteristics The normal curve is bell-shaped. The normal distribution is symmetric about its mean(50% of all values lie either site of the mean). Mean, Median, and Mode are equal

The Standardized Normal  Any normal distribution can be transformed into the standardized normal distribution(Z).  The standardized normal distribution has 0 mean and 1 standard deviation.  Any normal distribution can be converted into the standardized normal distribution by subtracting the mean from each data point and dividing by the standard deviation. The result from this operation is called z value or z score.

 The standard normal value=Z value=(x- µ)/ơ  Where  X is any data value  µ is mean of distribution  Ơ standard deviation Example: The monthly incomes of managers normally distributed with a mean of 1200$ and a standard deviation of 150$. What is the z value for an income X of 1400$? X of 1000$?

Solution: the z value can be computed by using the formula: Z value=(x- µ)/ ơ For x=1400: z=( )/150 =1.33 For x=1000: z=( )/150 =-1.33 The z value of 1.33 means that the monthly income of 1400$ for a manager is 1.33 standard deviation above the mean, and z value of shows that the monthly income of 1000$ is standard deviation below the mean.

The Probability under the normal curve  By computing the z value, we can determine the area or the probability under the standard normal curve by referring to the standard normal distribution table.  The total area under the curve is 1.0 Example: If z=2.84, what is the area between the z value and the mean? By looking at the z column and moving vertically down to z value of 2.8 and then looking at the z row and moving horizontally right to z value of It is This is the probability between the z value and the mean.

Example: Refer to the previous example. In that example the mean income is 1000$ and the standard deviation is 150$. a. What is the probability that a particular monthly income for a manager is between 800$ and 900$? b. What is the probability that income is more than 900$? c. What is the probability that the income is less than 800$?

Solution a.The probability of income between 800$ and 900$ For x=800: z value=( )/150=-2.66 For x=900: z value=( )/150=-2.0 The probability between the z value of and The mean, p(-2.66<=z<=0.0) is The Probability between the z value of -2.0 and The mean, p(-2.0<=z<=0.0) is

Therefore, the probability between the z value of -2.66(income=800$) and the z value of -2.0 (income=900$), p(-2.66<=z<=2.0) is the result of subtracting p(-2.66<=z<=0) with p(-2.0<=z<=0). p(-2.66<=z<=2.0)= p(-2.66<=z<=0)- p(-2.0<=z<=0) = = b. The probability that the income is more than 900$ (answer, p=0.9772) c. The probability that the income is less than 800$ (answer, p=0.0039)