Normal Curve Z-Scores Percentiles JV Stats Unit 4

What is a NORMAL DISTRIBUTION? Data that perfectly fills up a bell curve might be considered Normal. You will have a certain % of LOW Scores, a certain % of MEDIUM Scores, and a certain % of HIGH Scores. Test grades are a good example of what a Normal Dist might look like, you have a few really terrible scores, a few high scores, and most fall in the middle. The heights of ALL people in the world would probably be pretty close to NORMAL. Typically the more #’s you have, the closer they will become to a NORMAL Dist, think about all the tests you have ever taken in your life, they would probably form a Normal dist themselves. CAUTION!!....most REAL data is not perfectly NORMAL. Many things are APPROXIMATELY NORMAL. Be careful saying that something IS Normal.

Give 3 examples of data that would follow a Normal Dist. Heights of all women in the U.S. IQ Scores Attendence for each day at Disneyland in one year Give 3 examples of data that would NOT follow a Normal Dist. Salaries of professional baseball players Years of coins currently in the population Age of cars on the road

Standard Normal Curve N(0,1)…….Mean = 0 & StDev = 1

The Empirical Rule 68,95,99.7 68% of the data is within 1 StDev of the mean 95% of the data is within 2 StDev of the mean 99.7% of the data is within 3 StDev of the mean THIS IS TRUE ONLY IF THE DATA FOLLOWS A NORMAL DISTRIBUTION

Draw a Normal Curve with Mean μ = 70 and Standard Deviation σ = 10. Label your standard deviations What type of scores do you think these are from? Title your Normal Curve at the botttom 68% of the scores were between _____ & ______ What percent of the data was less than 80? What percent of the data was between 50 and 90? What score was at the 50th percentile? What score would be the 16th percentile? What score would be at the 84th percentile? 97.5% of all scores were above _____. What percent of the scores were between 50 & 80? What is the median score?

Test #3 Class Scores 74,64,64,60,74,72,74,58,84,74,58,68,80, 58,84,72,68,92,72,74,80,73,65,72,74,90, 82,86,88,64,78,68,76,72,52,52,58,62,88,76,66,72,84,64,64,89,90,68,62,81,80 Create a stemplot of this data (make sure to split your stems) DOES THESE TEST SCORES FOLLOW A NORMAL DISTRIBUTION? COMPARE TO THE EMPIRICAL RULE

Thanks To: Kiara Wilcox

The mean of the data is____ and the standard deviation is_____. How well do these scores fit into the Normal Condition according to the Empircal Rule? 73 11

The mean and median are pretty close The mean and median are pretty close. This is some evidence that the scores might be approximately Normal……but more evidence is needed.

Does this data follow a Normal Distribution? Mean = 2.55 StDev = .16 Median = 2.53 n = 122 Check the shape Check Mean & Med. Empirical Rule(68,95,99.7)

Compare to the Empirical Rule (68,95,99.7)

Sudoku Results

The distribution of Sudoku scores looks fairly symmetric

N(547,180) We will assume that the data is approximately Normal All times were converted into seconds. Sketch a Normal curve and make all appropriate labeling. Draw a vertical dashed line for your score, if you don’t know your score I will give you one.

Z - Scores A Z-score is the number of Standard Deviations above or below the mean

Calculate Your Score? Your score should be entered into the equation for X. Your Z-score tells you how you did compared to the mean score. Usually you want to have a positive Z-score….but since we were finishing a sudoku puzzle for time, you want a lower score.

Assume a Normal Distribution with a Mean μ = 70 and Standard Deviation σ = 10. Calculate the Z-score for a score of 83. Label the above Z-scores on a normal curve and estimate what percentile they would be at.

Finding the exact percentile Lets find the exact percentile for 83. Use your z-score and the table

This means that a score of 83 is at the 90.32 percentile A score of 83 gave us a Z-score of 1.30 This table ALWAYS gives the % of data to the LEFT on the Normal curve. This means that a score of 83 is at the 90.32 percentile

What % of the data are between the scores of 60 and 83? 60 has a z-score of -1 which gives .1587 to the left 83 we know has .9032 to the left Subtract…(.9032 - .1587) gives .7445

Mr. Pines gave a quiz last week to his AP Stats classes Mr. Pines gave a quiz last week to his AP Stats classes. The scores on the quiz followed a Normal distribution with a mean μ = 15. 1. One of his students, Nina scored a 12 on the quiz. Her Z-score was -1.5. What was the standard deviation of the scores?

3. What is her exact percentile? 2. Draw a Normal Curve for this distribution and estimate Nina’s percentile in the class on this quiz. 3. What is her exact percentile?

4. Another student Tony had a z-score of -2 4. Another student Tony had a z-score of -2.03, what percentile is his score at?

5. What quiz score was at the 79th percentile?

6. What percent of scores were between z-scores of -1.13 and 2.19?

Create your own Normal Distribution Make up a situation Give a mean Give a Standard Deviation Draw a Normal Curve Write 10 questions with answers(finding z-cores, finding percentiles, etc) Create your own Normal Distribution

Body Temperatures

Most people think that the “normal” adult body temperature is 98. 6°F Most people think that the “normal” adult body temperature is 98.6°F. That figure, based on a 19th-century study, has recently been challenged. In a 1992 article in the Journal of the American Medical Association, researchers reported that a more accurate figure may be 98.2°F. Furthermore, the standard deviation appeared to be around 0.7°F. Assume that a Normal model is appropriate. What fraction of people would be expected to have body temperatures above 98.6°F? What % of temperatures would be between 97.3°F and 100.2°F? What temperature would be at the 92nd percentile? DRAW A NORMAL CURVE FOR EACH PROBLEM

California Earthquakes

California Earthquakes since 1680. 4:1 stands for 4.1

What was the median magnitude of these earthquakes?(use your stemplot) There are 72 recorded earthquakes on this list What was the median magnitude of these earthquakes?(use your stemplot) Draw out a Normal Curve for this situation using 6.29 as the mean and .86 as the standard deviation. Make sure to label all your z-score lines. 68% of the earthquake magnitudes should be between ______ & ________? What is the 84th percentile for earthquakes magnitudes? 6.5 5.43 7.15 7.15

What is the percent of earthquakes have been less than 5.4? What percent of earthquakes have been between a 5.0 and a 6.8? What is the probability that the next earthquake we have is greater than an 5.1? 9. What is the probability that the next earthquake we have is greater than an 8.0? DRAW A NORMAL CURVE FOR EACH PROBLEM

10. What magnitude would be an earthquake at the 14th percentile? 11. 23% of earthquakes in California are over what magnitude? DRAW A NORMAL CURVE FOR EACH PROBLEM

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#11

On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many balls. When he hits his driver, the distance the ball travels follows a Normal distribution with mean 304 yards and standard deviation 8 yards. Tiger Woods

Sketch a Normal curve labeling 3 standard deviations above and below the mean. What percent of Tigers drives are between 296 and 312 yards? 95% of Tigers drives are between ______&______? What percent of Tigers drives were less than 320 yards? What percent of Tigers drives were over 288 yards? The 76th percentile of Tigers drives is what z-score? The 76th percentile of Tigers drives is what distance? What percent of Tigers drives are over 300 yards? What percent of Tigers drives are less than 300 yards? What percent of Tigers drives are between 300 and 310 yards? The Yellow problems can be solved using the Empirical Rule The White problems require the table and their own Normal Curve

CAUTION!! Not all data follows a Normal Distribution Data with outliers or skewness may not be Normally distributed Large samples will be closer to a Normal distribution than small samples Real life data is almost NEVER EXACTLY NORMAL.

COFFEE

The amount of coffee that Mr The amount of coffee that Mr. Pines drinks on a school day follows a Normal Distribution with a mean of 23.4 ounces and a standard deviation of 7.2 ounces.

N(23.4,7.2) Mr. Pines coffee drinking 1) Sketch a Normal curve labeling all important information for this situation 2) How often does Mr. Pines drink less than 20 ounces of coffee in a day? 3) How often does he drink between 15 and 30 ounces of coffee in a day? 4) 60% of the time he drinks more than ____ ounces of coffee in a day? 5) How many ounces of coffee would be considered outliers for Mr. Pines?

N(23.4,7.2) Mr. Pines coffee drinking 1) Sketch a Normal curve labeling all important information for this situation

N(23.4,7.2) Mr. Pines coffee drinking 2) How often does Mr. Pines drink less than 20 ounces of coffee in a day?

N(23.4,7.2) Mr. Pines coffee drinking 3) How often does he drink between 15 and 30 ounces of coffee in a day?

N(23.4,7.2) Mr. Pines coffee drinking 4) 60% of the time he drinks more than ____ ounces of coffee in a day?

N(23.4,7.2) Mr. Pines coffee drinking 5) How many ounces of coffee would be considered outliers for Mr. Pines?