Normal Probability Distributions

Intro to Normal Distributions & the STANDARD Normal Distribution

 1. mean, median, and mode are equal  2. bell shaped and symmetric about the mean  3. total area under the curve is 1  4. the curve approaches, but never touches the x axis as it extends away from the mean

 A Normal distribution with  mean = 0 and  standard deviation = 1

 The cumulative area is close to 0 for z-scores close to z =  The cumulative area increases as the z-score increases.  The cumulative area for z = 0 is  The cumulative area is close to 1 for z-scores close to z = 3.49

1. Sketch the curve and shade the appropriate area under the curve. 2. To find the area LEFT of z, find the area that corresponds to z on the table. 3. To find the area RIGHT of z, find the area that corresponds to z on the table, then subtract from To find the are BETWEEN 2 z-scores, find each area, then subtract.

 1. left of z =  2. right of z = 2.30  3. left of z = 0.45  4. right of z =  5. between z = and z = 1.89

Normal Distributions: Finding Probabilities

 1. Sketch a normal curve.  2. Find the z-score Recall: z = x - µ σ  3. Plot your z-score and shade as indicated.  4. Find the area under the curve using the standard normal distribution table (table #4)  5. Interpret the results to answer the question.

Find P(x < 200) Find P(x > 155) Find P(172 < x < 192)

 13. A survey was conducted to measure the heights of US men. In the survey, respondents were grouped by age. In the age group, the heights were normally distributed, with a mean of 69.9 inches and a standard deviation of 3.0 inches. A study participant is randomly selected. Find each prob:  A) his height is less than 66 in.  B) his height is between 66 and 72 in.  C) his height is more than 72 in.

 20. The times per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. An athlete is randomly selected. Find each probability.  A) the athlete uses a stairclimber for less than 17 minutes.  B) the athlete uses a stairclimber between 20 and 28 minutes.  C) the athlete uses a stairclimber for more than 30 minutes.

 26. Monthly utility bills are normally distributed with a mean of $100 and a standard deviation of $12.  A) What percent of the utility bills are more than $125?  B) If 300 utility bills are randomly selected, about how many would you expect to be less than $90?