PSUnit_II_Lesson_4_Determining_Probabilities (1).pptx

Determining_Probabilities

1. Determining Probabilities

2. Lesson Objectives At the end of this lesson, you are expected to: find areas between paired z-scores; find probabilities for the standard normal random variable z; and express areas under the normal curve using probability notation.

3. Pre-Assessment

4. Lesson Introduction S tandard normal distribution is a normal distribution with μ = 0 and σ = 1. A random variable with a standard normal distribution, denoted by X, is called a standard normal random variable. P robabilities associated with the standard normal random variables can be shown as areas under the standard normal curve.

5. Discussion Points Probability Notations Under the Normal Curve The following notations for a random variable are used in our various solutions concerning the normal curve. denotes the probability that the z-score is between a and b . denotes the probability that the z-score is greater than a . denotes the probability that the z-score is less than a . where a and b are z-score values.  

6. Discussion Points T o find the area of the region between z = 1 and z = 2, we subtract .3413 from .4772 resulting in .1359. It is graphically shown below.

7. Discussion Points T he regions under the normal curve in terms of percent, the graph of the distribution would look like this:

8. Discussion Points

9. Case 1: The required area, as depicted by the shaded regions under the curve in figure 2.8 and 2.9, are: ‘greater than z’ ‘at least z’ ‘more than z’ ‘to the right of z’ ‘above z’

10. Case 2: The required area is: ‘less than z’ ‘at most z’ ‘not more than z’ ‘not greater than z’ ‘to the left of z’

11. Case 3: Models when the required area is  

12. Example 1 Find the proportion of the area above z = –1. STEPS SOLUTION Draw a normal curve. Locate the z-value. Draw a line through the z-value Shade the required region. Consult the z-Table and find the area that corresponds to z = –1. z = –1 corresponds to an area of .3413 Examine the graph and use probability notation to form an equation showing the appropriate operation to get the required area. The graph suggests addition. The required area is equal to 0.3413 + 0.5 = 0.8413. That is, P(z > –1) = 0.3413 + 0.5 = 0.8413 Make a statement indicating the required area. The proportion of the area above is .8413.

13. Example 2 Find the area to the left of z = –1.5. STEPS SOLUTION Draw a normal curve. Locate the z-value. Draw a line through the z-value Shade the required region. Consult the z-Table and find the area that corresponds to z = –1. 5 z = 1.5 corresponds to the area 0.4332 Examine the graph and use probability notation to form an equation showing the appropriate operation to get the required area. The graph suggests subtraction. The required area is equal to 0.5 – 0.4332 = 0.0668 That is, P (z < –1.5) = 0.5 – 0.4332 = 0.0668 Make a statement indicating the required area. The proportion of the area to the left of z = –1.5 is 0.0668.

14. Example 3 Find the area between z = –2 and z = –1.5. STEPS SOLUTION Draw a normal curve. Locate the z-value. Draw a line through the z-value Shade the required region. Consult the z-Table and find the area that corresponds to z = –1. 5 z = –2 corresponds to .4772 z = –1.5 corresponds to .4332. Examine the graph and use probability notation to form an equation showing the appropriate operation to get the required area. The graph suggests subtraction. The required area is equal to 0.4772 – 0.4332 That is, P (–2 < z < –1.5) = 0.4772 – 0.4332 = 0.0440 Make a statement indicating the required area . The required area between z = –2 and z = –1.5 is 0.0440.

15. Exercise 1 1. Find the area greater than 2. Find the area above 3. Find the area between 4. Find the area between  

16. Exercises Find the area between z = –1.32 and z = 2.37. Complete the table below.

17. Problem Solving: 1. A brisk walk at 4 miles per hour burns an average of 300 calories per hour.If the standard deviation of the distribution is 8 calories, find the probability that a person who walks one hour at the rate of 4 miles per hour will burn the following calories. Assume the variable to be normally distributed. a) more than 280 calories c) between 278 and 318 calories b) less than 294 calories 2. If the systolic blood pressure for a certain group of obese people has a mean of 132 and a standard deviation of 8, find the probability that a randomly selected person will have the following blood pressure. Assume the variable is normally distributed. a. above 130 b. below 140 c. between 131 and 136

18. 3. For a medical study, a researcher wishes to select people in the middle of 40% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower reading that would qualify people to participate in the study. 4. If the scores for the test have a mean of 100 and a standard deviation of 15, find the percentage of scores that will fall below 115.

19. The Central Limit Theorem The Central Limit Theorem states that if you take sufficiently large samples from a population, the samples’ means will be normally distributed, even if the population isn’t normally distributed. It states further that as the sample size increases, the shape of the distribution of the sample means taken from a population with mean µ and standard deviation σ will approach a normal distribution. This distribution will have mean µ and a standard deviation . The Central Limit Theorem can be used to answer questions about sample means in the same manner that the normal distribution can be used to answer about individual values. There is a new formula to be used for the z values: z = where ; is the sample mean and is the standard deviation  

20. It is important to remember two things when using the central limit theorem: When the original variable is normally distributed, the distribution of the sample means will be normally distributed, for any sample size n. When the distribution of the original variable departs from normality, a sample size of 30 or more is needed to use the normal distribution to approximate the distribution of the sample means. The larger the sample, the better the approximation will be. Sample problems where CLT is applied : 1. The mean serum cholesterol of a large population of overweight adults is 220 mg/dl and the standard deviation is 16.3 mg/dl. If a sample of 30 adults is selected, find the probability that the mean will be between 220 and 222 mg/dl. 2. The average age of accountants is 43 years, and with a standard deviation of 5 years. If an accountant firm employs 30 accountants, find the probability that the average age of the group is greater than 44.2 years old.

21. 3. The average annual precipitation for a certain city is 30.83 inches, with a standard deviation of 5 inches. If a random sample of 10 years is selected, find the probability that the mean will be between 32 and 33 inches. Assume that the variable is normally distributed. 4. The mean weight of a 20-year-old females is 126 pounds and the standard deviation is 15.7. If a sample of 25 females is selected, find the probability that the mean of the sample will be greater than 128.3 pounds. Assume that the variable is normally distributed.

22. Solve the following problems. Show complete solutions. 1. The average life of a brand of automobile tires is 30,000 miles, with a standard deviation of 2,000 miles. If a tire is selected and tested, find the probability that will have the following lifetime. Assume the variable is normally distributed. a) between 25,000 and 29,000 2. For a medical study, a researcher wishes to select people in the middle 75% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower readings that would qualify people to participate in the study. 3. The average number of kilos of meat a person consumes in a year is 100 kg. Assume that the standard deviation is 11 kg and the distribution is approximately normal. a. Find the probability that a person selected at random consumes less than 105 kg per year. b. If a sample of 40 individuals is selected, find the probability that the mean of the sample will be less than 105 kg per year.

23. Summary Steps in Determining Areas Under the Normal Curve Use a cardboard model to draw a normal curve. Locate the given z-value or values at the base line. Draw a vertical line through these values. Shade the required region. Find models, if any. Consult the z-Table to find the areas that correspond to the given z-value or values. Examine the graph and use probability notation to form an equation showing an appropriate operation to get the required area. Make a statement indicating the required area.