Distribution........................pptx

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1. NORMAL DISTRIBUTION

2. Learning Competency: The learner illustrates a random variable and its characteristics.

3. Normal Distribution It is the cornerstone of the application of statistical inference in analysis of data because the distributions of several important sample statistics tend towards a normal distribution as the sample size increases.

4. Normal Random Variable The normal random variable is a continuous random variable that follows the normal distribution with mean and standard deviation .  

5. Normal Equation It is the probability density function of the normal distribution. The value of the random variable Y is: where x is the normal random variable.  

6. Normal Curve The graph of a normal distribution is bell-shaped. It depends on two factors: the mean and standard deviation. The mean determines the center of the graph and the standard deviation determines its height and width. Normal distributions with higher standard deviation create curves with smaller height and bigger width.

7. Normal Distribution It is a representative of continuous data (such as measurements) that have some variation but no bias above or below the mean.

8. Characteristics of the Normal Probability Distribution The curve has a single peak. It is bell-shaped or shape like a Mexican sombrero. The mean (average) lies at the center of the distribution. The distribution is symmetrical about the mean. The two tails extend indefinitely in both directions coming closer and closer to the horizontal axis but never quite touching it. It includes 100% of the data, so the area under the curve is 1.

9. Empirical Rule It is a rule that use standard deviation so that we can measure with great precision percentage of items that fall within specific ranges under a normal distribution.

10. Empirical Rule 50% of all data points are above the mean and 50% are below 68.27% of all data points are within 1 standard deviation of the mean 95.45% of all data points are within 2 standard deviations of the mean Approximately 99.73% of all data points are within 3 standard deviations of the mean

11. AREAS UNDER A NORMAL CURVE AND THE Z SCORE

12. Learning Competencies: At the end of the lesson, learners are expected to: identify regions under the normal curve convert a normal random variable to a standard normal variable and vice versa find the area of z under the normal curve

13. Normal Distribution It is the most common distribution that applies to many real life data. Examples: heights of people weights of people exam scores blood pressure

14. Characteristics of the Normal Probability Distribution The curve has a single peak. It is bell-shaped or shape like a Mexican sombrero. The mean (average) lies at the center of the distribution. The distribution is symmetrical about the mean. The two tails extend indefinitely in both directions coming closer and closer to the horizontal axis but never quite touching it. It includes 100% of the data, so the area under the curve is 1.

15. Standard Score or Z-Score A z-score also called a standard score gives an idea of how far from the mean a data point is. It’s measure of how many standard deviations below or above the population mean a raw score is.

16. Standard Score or Z-Score It allows us to calculate the probability of a score occurring within our normal distribution. It enables us to compare two scores that are from different normal distributions.

17. Converting to a Standard Normal Variable Z and Vice Versa To convert a value of a normal random variable x to its standard normal variable Z (or Z score value), we use the formula: where: Z = the number of standard deviations from x to the mean of the distribution x = the value of the random variable = standard deviation of the distribution = the mean of the distribution of the random variable  

18. Converting to a Standard Normal Variable Z and Vice Versa To convert a standard normal variable Z (or Z score) into its normal random variable x value, we use the formula: where: Z = the number of standard deviations from x to the mean of the distribution x = the value of the random variable = standard deviation of the distribution = the mean of the distribution of the random variable  

19. Example 1 Suppose you have a set of test scores that are normally distributed with mean equal to 80 and standard deviation equal to 5. If you got 75, what is your z-score?

20. Example 2 Math exam scores are normally distributed with a mean of 80 and a standard deviation of 5. If you got a z-score of -2, what is your exam score?

21. Example 3 You would like to compare your performance during the summative tests in HOPE 2 and StatProb subjects. You are enrolled in a class with 44 students. With the given table below, where did you perform better? Subjects X HOPE 2 38 37.5 1.5 StatProb 36 32.5 6 Subjects X HOPE 2 38 37.5 1.5 StatProb 36 32.5 6

22. ACTIVITY #11 Solve the unknown.  

23. Finding the Area Under a Normal Curve Given a Z Value A specific proportion of the area of the region under the curve can be calculated manually using the formula where: Y = represents the height of the curve at a particular value of X = represents any score in the distribution = represents the standard deviation of the population = represents the population mean = 3.1416 = 2.7183  

24. Steps in Finding the Area (or Area Percentage) Under a Normal Curve Given a Z Value Express the given z-value into a three-digit number. Using the z-Table, find the first 2 digits on the first column. Find the third digit on the first row on the right. Read the area for probability at the intersection of the row (first 2 digit number) and column (third digit number). The value observed at the intersection indicates the area of the given z-value.

25. AREAS UNDER THE NORMAL CURVE

26. Example 1 Find the area that corresponds to a z-score of 1.5. (Look for 1.5 on the first column and 0 on the first row) A = 0.93319

27. Example 2 Find the area that corresponds to a z-score of 1.09. (Look for 1 on the first column and 0.09 on the first row) A = 0.86214

28. Example 3 Find the area that corresponds to a z-score of 0.27. (Look for 0.2 on the first column and 0.07 on the first row) A = 0.60642

29. Example 4 Find the area of the region corresponding to the z-score = -1. (Look for -1 on the first column and 0 on the first row) A = 0.15866

30. Example 5 Find the area of the region corresponding to the z-score = -2.36. (Look for -2.3 on the first column and 0.06 on the first row) A = 0.00914

31. ACTIVITY #12 Find the area under the normal curve given the z-value. 1. 2 6. -2.24 2. -3 7. 0.4 3. 1.2 8. -0.5 4. -2.1 9. 0.67 5. 1.52 10. -0.48

32. SHADED REGION UNDER THE NORMAL CURVE

33. Learning Competencies: At the end of the lesson, learners are expected to: find the shaded region under the normal curve given specific conditions using the standard normal table define percentile compute probabilities and percentiles using the standard normal table

34. Standard Normal Distribution It is a normal distribution with a mean of zero and standard deviation of one and has a total area under its normal curve of one.  

35. Probability Notations P(z<a) P(z>a) P(a<z<b) The probability that the value is… “less than z” “to the left of z” “below z” “lower than z” “under z” The probability that the value is… “greater than z” “to the right of z” “above z” “more than z” “at least z” The probability that the z value is between a and b.

36. Steps in Finding the Areas Under the Normal Curve Compute for the z-score for observation x. Draw the Normal Distribution Curve and shade the area. Find the corresponding area of the z-score.

37. Example 1 Vehicles’ speed at McArthur Hi-way have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What is the probability that a randomly selected car is going 73 mph or less?

38. Example 2 The pulse rates of adult females have a normal curve distribution with a mean of 75 beats per minute (bpm) and a standard deviation of 8 bpm. Find the probability that a randomly selected female has a pulse rate greater than 85 bpm.

39. Example 3 The mean for IQ scores is 100 and the standard deviation is 15. What proportion of IQ scores falls between 100 and 130?

40. Example 4 Find the area of the region between the and under the standard normal curve.  

41. Example 5 Compute for the probability that the Z may take the value/s. Greater than 2 Less than 2 Between 1 and 2

42. ACTIVITY #13 Find the area under the normal curve in each of the following cases. 1. Less than z = 2.35 6. Greater than z = 0.89 2. Less than z = -1.31 7. Between z = 0 and z = 1.63 3. Less than z = 0.35 8. Between z = 0 and z = -1.78 4. Greater than z = 1.85 9. Between z = 1.56 and 2.51 5. Greater than z = -0.95 10. Between z = -2.76 and z = -1.25

43. PERCENTILES UNDER THE NORMAL CURVE

44. Percentile It is a measure used in statistics indicating the value below which a given percentage of observation in a group of observation falls. It is a measure of relative standing as it measures the relationship of a measurement of the rest of the data.

45. Steps in Finding the Percentiles Under the Normal Curve Draw the Normal Distribution Curve and shade the area. Express percentile as probability. Refer to the table of areas under the normal curve. Locate its the corresponding area. Locate the nearest value. Find the x-score that lies in the percentile, using the formula . Locate the z-value and draw a line through under the curve. Describe the shaded region.  

46. Example 1 Find the 95 th percentile of the normal curve.

47. Example 2 What is the 33 rd percentile for the intelligence Quotient scores when the mean and standard deviation  

48. Example 3 Consider the normal distribution of IQs with a mean of 100 and a standard deviation of 25. What percentage of IQs are: greater than 95? less than 120? between 90 and 110?

49. Example 4 In an English test, the mean is 60 and the standard deviation is 6. Assuming the scores are normally distributed, what percent of the score is: greater than 65? less than 70? between 50 and 65?