Lesson Objectives: In this chapter you will learn about the properties of density curves, particularly that of normal curves. You will also learn how to calculate the proportion of observations that fall within a specific interval of a normal distribution given the population mean and standard deviation (68-95-99.7 Rule). We will also review how to assess whether a given data set is normally distributed or not since you can only use normal calculations if the distribution is normally distributed. The calculations you learn in this chapter will be used again when we begin our study of statistical inference (using sample data to estimate or test population characteristics).

Date Topics Objectives: Students will be able to… Assignment

Feb 2

2.1 Transforming Data, Density Curves

• Describe the effect of adding, subtracting, multiplying by, or dividing by a constant on the shape, center, and spread of a distribution of data.

• Approximately locate the median (equal-areas point) and the mean (balance point) on a density curve.

TPS: Read pg. 78 – 83 Watch Video #1; Video #2 Take notes. Complete pg. 2 – 3 of Chapter 2 Packet [We will work on page 4 -5 in class]

2.2 Normal Distributions, The 68-95-99.7 Rule, The Standard Normal Distribution

• Use the 68–95–99.7 rule to estimate the percent of observations from a Normal distribution that fall in an interval involving points one, two, or three standard deviations on either side of the mean.

• Use the standard Normal distribution to calculate the proportion of values in a specified interval.

• Use the standard Normal distribution to determine a z-score from a percentile.

TPS: Read pg. 85 -88, 90 Watch Video #3. Take notes. Complete pg. 6 -7 of Chapter 2 Packet

Feb 5

2.2 Normal Distribution Calculations

• Use Table A to find the percentile of a value from any Normal distribution and the value that corresponds to a given percentile.

TPS: Read pg. 93 – 95, 97 -103. Watch Video #4 – Take notes. Complete pg. 8 - 10 of Chapter 2 Packet

2.2 Assessing Normality

• Make an appropriate graph to determine if a distribution is bell-shaped.

• Use the 68-95-99.7 rule to assess Normality of a data set.

• Interpret a Normal probability plot

TPS: Read pg. 104 -109, 112. Watch Videos: Empirical Rule; Assessing Normality Take notes. Complete pg. 11 - 15 of Chapter 2 Packet

Chapter 2 Packet Due Feb. 6, 2018. Start Chapter 3 Packet.

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Measuring Position: Percentiles – [VIDEO #1]

One way to describe the location of a value in a distribution is to tell what percent of observations are _____ it.

On her first AP Stats exam, Jenny scored an 86%. What percentile she is in with respect to the rest of her class?

Context: ____________________________________________________

_____________________________________________________

____________________________________________________

Reading an Ogive

A histogram does a good job of displaying the distribution of values of a variable, but it doesn’t tell us about the

relative standing of an individual observation. If we want this information, then we need to construct a

_________________________________________________________________ often called an ___________.

1. What grade is the center of this distribution? _________

2. What grade corresponds to Q1? ___________

3. What grade corresponds to the 60th percentile? _________

4. What approx. percentile would a 77% put you in? ______

5. What minimum score is required to be in at least the 90th

percentile? ________

Class Count Cumulative Frequency

Relative Cumulative Frequency

65 – 69 70 – 74 75 – 79 80 – 84 85 – 89 90 – 94

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Standardizing Let’s stick with Jenny’s score of an 86% on her first test. Now, for the second test, she scored another 86%. Compared to her first score, she stayed the same obviously, but how does she compare with her peers the 2nd time around? We could compare the percentiles for each test to see which test Jenny really did better at compared to her peers, but what if you did NOT have all the data handy? What statistical values do you think we would need in order to determine Jenny’s performance? ________________________ __________________________ ________________________ __________________________

To standardize a value use the formula: z = -------------------------------

A standardized value is often called a z-score. A z-score tells us how many standard deviations the original

observation falls away from the mean, and in what direction.

1. Compute and compare Jenny’s z-scores for her two tests.

2. Compute and interpret Maxwell’s score of a 75% on the first test.

3. Compute and interpret Brittany’s score of a 95% on the second test.

4. Who did better? Jeff who scored a 79% on the first test or James who scored a 82% on the second test.

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Transforming Data – [See Effects of Transformations Video#2]

Let’s input this data into L1. Then, get out of the editor. Press STAT, go over to “CALC”, and select

“1-Var Stats”. Now, press 2ND, STAT, select L1, and hit ENTER for some results!!!

What changed? à (WHY???) ß What did not change?

What changed? à (WHY???) ß What did not change?

See if you can get each number without using “1-Var Stats” by applying the patterns that we have observed.

What do you think will change and how so? What do you think will not change?

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How is each summary statistic of x affected by the linear transformation 𝑥!"# = 𝑎 + 𝑏𝑥? Median!"# = ______________________________________ Mean!"# = ______________________________________ Range!"# = ______________________________________ IQR!"# = ______________________________________ Std.Dev!"# = ______________________________________ Variance!"# = ______________________________________ Suppose a teacher gave a test for which 𝑥 = 70, and 𝑠 = 21. He wants to apply a linear transformation 𝑥!"# = 𝑎 + 𝑏𝑥 to “scale” the grades so that 𝑥!"# = 82 and 𝑠!"# = 7. Find a and b.

1. What are the effects of Linear Transformations on a data set? a. What would happen to the measures of center (𝑥 and M) and the measures of spread (range, IQR, and s)

if a certain value were added to or subtracted from each piece of data?

b. What would happen to the measures of center (𝑥 and M) and the measures of spread (range, IQR, and s) if each piece of data was multiplied or divided by a certain value?

c. Do linear transformations change the shape of a distribution?

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Density Curves – [VIDEO #3]

The Median and Mean of a Density Curve: 1. Draw a left-skewed, symmetric, and right skewed density curve. 2. Label the location of the mean and median on each.

3. The mean on a density curve is located at the ________________________.

4. The median on a density curve is at the __________________________________.

5. If the mean and median are equal then the density curve is ____________________.

6. What must the total area under the curve equal? ______

Find the proportion of observations within each interval: 7. 0 ≤ X ≤ 0.2

8. X ≥ .4

9. 0.1 ≤ X ≤ 1.25

10. Use the density curve shown to answer the following questions. What proportion of observations fall within each interval? a. 0 ≤ X ≤ 1

b. X ≤ ½

c. X ≥ ½

d. ½ ≤ X ≤ 1½

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

5

4

3

2

1

A Density Curve is a curve that • Is always on or above the horizontal axis • Has an area of exactly 1 underneath it.

A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range.

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The Normal Distribution Many real-life distributions follow a particular pattern. The pattern that arises for many data sets is one consisting of a single peak that is symmetrical, and has a bell-shape. We call this special distribution the normal distribution. What are a few examples of variables that are approximately normally distributed?

1. ______________________________

2. ______________________________

3. ______________________________

From a To a Normal Histogram Distribution Note: The Normal Distribution is an idealized approximation to the histogram. A normal curve is a specific type of density curve. Let’s compare the two!

A Density Curve A Normal Curve

Example(s):

Is it always on or above the x-axis?

Does it have an area of exactly 1 beneath it?

Does the shaded area represent the proportion of values in that region?

Is it bell shaped?

Does it have a single peak?

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Normal Distributions, Continued – [VIDEO #4]

• All normal distributions (normal curves) have the same overall shape.

• They are all _______________, single-_________, and _______ shaped.

• A specific normal curve is described by stating 2 features:

1. Its ______________ (µ) and

2. Its _________________ ________________ (σ).

• The mean is located at the ________________ of the symmetric curve, and is the same as the

___________________.

Normal Curve #1 Normal Curve #2

• What would happen to the following two curves if you increased:

a. µ, but not σ?

b. σ, but not µ?

c. Which has a larger standard deviation?

d. How many different normal curves are there?

e. What is the shorthand notation for normal curves?

Standard Normal Distributions

The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. The shorthand notation for the standard Normal distribution is __________.

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The 68-95-99.7 Rule

Although there are many normal curves, they all have common properties. In particular, all normal distributions obey the following rule.

• 68% of the observations fall within ___σ of the mean (µ)

• 95% of the observations fall within ___σ of the mean (µ)

• 99.7% of the observations fall within ___σ of the mean (µ)

1. The blood cholesterol readings of a group of women follow the N(172, 14) distribution.

a. Within what range do about 95% of the readings fall?

b. About what percent of women have readings between 158 and 172?

c. Readings higher than 200 are undesirable. About what percent of the readings are undesirable?

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2. The amount of juice dispensed from a machine is normally distributed with a mean of 10.5 oz and a standard deviation of 0.75 oz.

a. About 68% of the amounts dispensed fall within what range?

b. About what percent of the time will the machine overfill the 12 oz cup?

c. A beverage that contained 9 oz of juice falls in what percentile?

Normal Distribution Calculations: There are two efficient ways to solve a normal calculation problem:

a. 68-95-99.7 Rule

b. Calculator (Normalcdf or Invnorm)

c. Using standardized scores with Table A

Practice Problems: 1. Megan scored 700 on the math portion of the

SAT. Math SAT scores are distributed according to the N(500,100) distribution. In what percentile did she fall?

2. Justin stayed up late the night before and scored a 480. Math SAT scores are distributed according to the N(500,100) distribution. What is his percentile?

3. Heather scored 2.25 𝜎 above the mean. Math SAT scores are distributed according to the N(500,100) distribution. What is her percentile?

4. What percentage of people score higher than 550?

5. What percent of scores fall between 650 and 800?

6. What score would you have to get to be in the 95th percentile?

7. How high must a student score to be in the top 10% of all students taking the SAT’s?

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The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of cholesterol levels for 14-year old boys is normally distributed with a mean of µ = 170 mg per deciliter of blood with a standard deviation of σ = 30 mg/dl. Levels above 240 mg/dl may require medical attention. 8. What percent of 14-year old boys have more than

240 mg/dl of cholesterol? 9. What percent of 14-year old boys have between 170

and 240 mg/dl? 10. What percent of 14-year old boys have less than 100

mg/dl? 11. What cholesterol level puts a boy in the top 5% of

all 14-year old boys?

Calculate the proportion of observations from a standard normal distribution that falls in each of the following regions. In each case, shade the area representing the region. 12. z ≤ -1.25 13. z ≥ -1.25 14. z > 1.75 15. –1.25 < z < 1.75

Calculate the value, z, of a standard normal curve that satisfies each of the following conditions.

16. The point z, with 10% of the observations falling below it.

17. The point z, with 20% of the observations falling above it.

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18. The number z, such that the proportion of observations that are less than z is 0.75.

19. The number z, such that 5% of all observations are

greater than z.

A company pays its employees an average wage of $9.25 an hour with a standard deviation of 60 cents. If the wages are approximately normally distributed determine: 20. What percent of workers earn more than $12/hr?

21. What percent of workers earn between $8.65 and

$9.85 per hour?

22. How much money do the top 5% of workers earn?

The quartiles of any density curve are the points with area 0.25 and 0.75 to the left under the curve. 23. What are the quartiles of a normal distribution?

24. What are the quartiles for the salaries earned by the workers in the example above?

Answer the following questions using the 68-95-99.7 rule. Grocery shoppers spend, on average, $75 per shopping trip with a standard deviation of $25. 25. What percent of shoppers spend more than $125 on a

single grocery-shopping trip?

26. How much money do the middle 95% of shoppers typically spend?

27. How much money does a person spend if they are

at the 84th percentile?

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The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are normally distributed with µ = 100 and s = 15. 1. What score would represent the 50th percentile?

Explain. 2. Approximately what percent of the scores fall in the

range from 70 to 130? 3. A score in what range would represent the top 16% of

the scores? 4. In a study of elite distance runners, the mean weight

was reported to be 115 pounds, with a standard deviation of 4 pounds. Assuming that the distribution of weights is normal, sketch the density curve of the weight distribution, with the horizontal axis marked in pounds. Find P(X < 107).

5. Jill scores 780 on the mathematics part of the SAT.

The distribution of SAT scores in a reference population is normally distributed with mean 500 and standard deviation 100. Jack takes the ACT mathematics test and scores 32. ACT scores are normally distributed with a mean of 18 and a standard deviation of 6. Find the standardized scores for both students.

6. Assuming that both tests measure the same kind of ability, who has the higher score, and why?

Runner’s World reports that the times of the finishers in the New York City 10-km run are normally distributed with a mean of 61 minutes and a standard deviation of 9 minutes. 1. Find the proportion of runners who take more

than 70 minutes to finish. Draw a sketch to show this proportion.

2. Find the proportion of runners who finish in less

than 43 minutes. Draw a sketch to show this proportion.

The Chapin Social Insight Test evaluates how accurately the subject appraises other people. In the reference population used to develop the test, scores are approximately normally distributed with mean 25 and standard deviation 5. The range of possible scores is 0 to 41. 3. If a randomly selected student has a score of 40, then

how many standard deviations away from the mean is that student’s score?

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4. Determine the standardized value for the score of 22. The Graduate Record Examinations are widely used to help predict the performance of applicants to graduate schools. The range of possible scores on a GRE is 200 to 900. The psychology department at a university finds that the scores of its applicants on the quantitative GRE are approximately normal with mean = 544 and standard deviation = 103. Use the table or your calculator to find the relative frequency of applicants whose score X satisfies the following: (Be sure to draw a normal curve and shade the area under the curve that represents the answer to the question.) 5. X < 500 6. 500 < X < 700 7. What minimum score would a student need in order

to score better than 77% of those taking the test?

A lunch stand in the business district has a mean daily gross income of $420 with a standard deviation of $50. Assume that the daily gross income is normally distributed. 8. If a randomly selected day has a gross income of

$320, then how many standard deviations away from the mean is that day’s gross income?

9. Determine the standardized value for the daily

income of $520.

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AftercompletingChapter2,youshouldknow:

q The properties of a normal curve.

q How to read and interpret an ogive

q How to apply linear transformations to sets of data.

q The properties of a density curve. (Remember, the density curve only has 2 properties, it doesn’t have

to be bell shaped like the normal curve, and it may not be a “curve” at all).

q The properties of a standard normal curve (Exactly symmetrical, mean= 0, standard deviation= 1)

q The properties of the standard deviation.

q Whether a curve is skewed left, right, symmetrical, or indeterminable based on a picture.

q How to estimate the mean, standard deviation, and the median based on a picture.

q How to perform the calculations of the normal distribution by the 68-95-99.7 rule, on the calculator

Normalcdf(lower bound, upper bound, mean, standard deviation), and by using Table A.

q How to sketch a normal curve and shade the appropriate areas(s).

q How to calculate a value given a certain percentile by the 68-95-99.7 rule, on the calculator.

InvNorm(percentile as a decimal, mean, standard deviation), and by using Table A.

q How to compare unlike quantities by calculating the z-score by using the calculator, and by using the

table.