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AP Statistics Monday, August 29, 2011 • Density Curves PowerPoint • HW: 2.9 – 2.13 • QUIZ THURSDAY!

AP STATS - Chapter 2 Density Curves and Normal Probability Distributions

Sampling Distribution (Histogram) and Density Curve (Red Curve)

Density Curves

A density curve describes the overall pattern of a distribution. We will use the smooth curve to describe what proportions of the observations fall in each range of values, not the counts of observations. Areas under the curve represent proportions of the observations. Total area under the curve is exactly 1.

Various Density Curves

Uniform Population Model

Total area under the curve (model) will always equal 1.

AP Statistics Tuesday, August 30, 2011 • 68 – 95 – 99.7% Rule • Example Worksheet • HW: Worksheet • QUIZ THURSDAY

What does a population that is normally distributed look like? = 80 and = 10

X

50

60

70

80

90

100

110

Empirical Rule 68-95-99.7% RULE

68%

95% 99.7%

Empirical Rule — restated 68% of the data values fall within 1 standard deviation of the mean in either direction 95% of the data values fall within 2 standard deviation of the mean in either direction 99.7% of the data values fall within 3 standard deviation of the mean in either direction Remember values in a data set must appear to be a normal bell-shaped histogram, dotplot, or stemplot to use the Empirical Rule!

Empirical Rule 34% 34% 68% 47.5%

47.5% 95%

49.85%

49.85%

99.7%

Average American adult male height is 69 inches (5’ 9”) tall with a standard deviation of 2.5 inches.

Empirical Rule-- Let H~N(69, 2.5) What is the likelihood that a randomly selected adult male would have a height less than 69 inches?

Answer: P(h < 69) = .50 P represents Probability

h represents one adult male height

AP Statistics Wednesday, August 31, 2011 • Review Worksheet • QUIZ TOMORROW • OPEN HOUSE TONIGHT

AP Statistics Thursday, September 1, 2011 • QUIZ

AP Statistics Friday, September 2, 2011 • Standard Normal Distribution (non calculator) – Number of finger taps in 1 minute – Calculate class mean and standard deviation – Read article about Pujols

• HW: 2.1, 2.3, 2.29, 2.31 – 2.33 • 3 DAY WEEKEND!!

What is a z-score? Standardized values based on the population mean (µ) and standard deviation (α).

Z ~ N (0, 1) # OF STANDARD DEVIATIONS ABOVE (+) OR BELOW (-) THE MEAN

z

x

Assuming X ~ N(66, 2), use the formula to calculate the corresponding z-scores for the x-values of 60, 62, 64, 66, 68, 70, and 72.

z

x

If we don’t know the values of x, but we know X ~ N(40, 4), then we can calculate the missing corresponding x-values when the z-score is –3, -2, -1, 0, 1, 2, and 3. Remember Z ~N(0, 1).

z

x

Example 1 Suppose the average height of freshmen at LHS is 60 inches with a standard deviation of 1.5 inches. What is the z-score for a freshman who has a height of a. 58 inches? b. 60.15 inches?

Example 2 Suppose the average height of sophomores at LHS is 62 inches with a standard deviation of 2 inches. What is the height of the sophomore (x-value) that corresponds to a a)z-score = 0? b)z-score = -2.44? c)z-score = 3.1?

Example 3 Suppose the average height of juniors at LHS is unknown but the standard deviation is 2.5 inches. What is the population mean height of juniors if a)a junior 66 inches tall corresponds to a z-score of -.75?

Example 4 Suppose the height of seniors at LHS is 67 inches but the standard deviation is unknown. What is the standard deviation knowing a)a senior 68.5 inches tall has a corresponding z-score of .87? b)a senior 63 inches tall has a corresponding z-score of –2.43? (This resulting population standard deviation should be different from the answer to a.)

Example 5 An incoming freshman took her college’s placement exams in French and math. In French, she scored 82 and in math, 86. The overall results on the French exam had a mean of 72 and a standard deviation of 8, while the mean math score was 76 with a standard deviation 12. On which exam did she do better RELATIVE to her classmates?

Using the z table (pink sheet) Find the following probabilities: 1. P(z < 1) 2. P(z < 0)

3. P(z < 1.5)

4. P(z > 1.5)

5. P(2.3 < z < 3.1)

Why do we use z-scores? To answer questions such as… 1. Suppose teachers at LHS have an age distribution X ~ N(40, 8). What is the likelihood that a randomly selected teacher from this population would have an age of 24 or younger?

Why do we use z-scores? 2. Assuming the same distribution exists for age of teachers at LHS, how likely is it for a randomly selected teacher from LHS to be older than 50 years of age?

Why do we use z-scores? 3. Again using the same teacher age distribution at LHS, what is the probability that a randomly selected teacher’s age would fall somewhere between 34 and 50 years of age?

Why do we use z-scores? 4. Again using the same teacher age distribution at LHS, at what age do 25% of the teachers fall below?

AP Statistics Tuesday, September 5, 2011 • Standard Normal Distribution (calculator) • Turn on / Link “Catalog Help” 2nd VARS (DIST) 2:normalcdf( 3:invNorm( • HW: Worksheet

(gives you a list of distributions we will use) lower bound, upper bound [, µ, α] area [, µ, α]

Area to the LEFT

Z-Scores on the CALCULATOR 1. Suppose teachers at LHS have an age distribution X ~ N(40, 8). What is the likelihood that a randomly selected teacher from this population would have an age of 24 or younger?

Z-Scores on the CALCULATOR 2. Assuming the same distribution exists for age of teachers at LHS, how likely is it for a randomly selected teacher from LHS to be older than 50 years of age?

Z-Scores on the CALCULATOR 3. Again using the same teacher age distribution at LHS, what is the probability that a randomly selected teacher’s age would fall somewhere between 34 and 50 years of age?

Z-Scores on the CALCULATOR 4. Again using the same teacher age distribution at LHS, at what age do 25% of the teachers fall below?

Z-Scores on the CALCULATOR 5. Again using the same teacher age distribution at LHS, at what age do 10% of the teachers fall above?

View more...
AP STATS - Chapter 2 Density Curves and Normal Probability Distributions

Sampling Distribution (Histogram) and Density Curve (Red Curve)

Density Curves

A density curve describes the overall pattern of a distribution. We will use the smooth curve to describe what proportions of the observations fall in each range of values, not the counts of observations. Areas under the curve represent proportions of the observations. Total area under the curve is exactly 1.

Various Density Curves

Uniform Population Model

Total area under the curve (model) will always equal 1.

AP Statistics Tuesday, August 30, 2011 • 68 – 95 – 99.7% Rule • Example Worksheet • HW: Worksheet • QUIZ THURSDAY

What does a population that is normally distributed look like? = 80 and = 10

X

50

60

70

80

90

100

110

Empirical Rule 68-95-99.7% RULE

68%

95% 99.7%

Empirical Rule — restated 68% of the data values fall within 1 standard deviation of the mean in either direction 95% of the data values fall within 2 standard deviation of the mean in either direction 99.7% of the data values fall within 3 standard deviation of the mean in either direction Remember values in a data set must appear to be a normal bell-shaped histogram, dotplot, or stemplot to use the Empirical Rule!

Empirical Rule 34% 34% 68% 47.5%

47.5% 95%

49.85%

49.85%

99.7%

Average American adult male height is 69 inches (5’ 9”) tall with a standard deviation of 2.5 inches.

Empirical Rule-- Let H~N(69, 2.5) What is the likelihood that a randomly selected adult male would have a height less than 69 inches?

Answer: P(h < 69) = .50 P represents Probability

h represents one adult male height

AP Statistics Wednesday, August 31, 2011 • Review Worksheet • QUIZ TOMORROW • OPEN HOUSE TONIGHT

AP Statistics Thursday, September 1, 2011 • QUIZ

AP Statistics Friday, September 2, 2011 • Standard Normal Distribution (non calculator) – Number of finger taps in 1 minute – Calculate class mean and standard deviation – Read article about Pujols

• HW: 2.1, 2.3, 2.29, 2.31 – 2.33 • 3 DAY WEEKEND!!

What is a z-score? Standardized values based on the population mean (µ) and standard deviation (α).

Z ~ N (0, 1) # OF STANDARD DEVIATIONS ABOVE (+) OR BELOW (-) THE MEAN

z

x

Assuming X ~ N(66, 2), use the formula to calculate the corresponding z-scores for the x-values of 60, 62, 64, 66, 68, 70, and 72.

z

x

If we don’t know the values of x, but we know X ~ N(40, 4), then we can calculate the missing corresponding x-values when the z-score is –3, -2, -1, 0, 1, 2, and 3. Remember Z ~N(0, 1).

z

x

Example 1 Suppose the average height of freshmen at LHS is 60 inches with a standard deviation of 1.5 inches. What is the z-score for a freshman who has a height of a. 58 inches? b. 60.15 inches?

Example 2 Suppose the average height of sophomores at LHS is 62 inches with a standard deviation of 2 inches. What is the height of the sophomore (x-value) that corresponds to a a)z-score = 0? b)z-score = -2.44? c)z-score = 3.1?

Example 3 Suppose the average height of juniors at LHS is unknown but the standard deviation is 2.5 inches. What is the population mean height of juniors if a)a junior 66 inches tall corresponds to a z-score of -.75?

Example 4 Suppose the height of seniors at LHS is 67 inches but the standard deviation is unknown. What is the standard deviation knowing a)a senior 68.5 inches tall has a corresponding z-score of .87? b)a senior 63 inches tall has a corresponding z-score of –2.43? (This resulting population standard deviation should be different from the answer to a.)

Example 5 An incoming freshman took her college’s placement exams in French and math. In French, she scored 82 and in math, 86. The overall results on the French exam had a mean of 72 and a standard deviation of 8, while the mean math score was 76 with a standard deviation 12. On which exam did she do better RELATIVE to her classmates?

Using the z table (pink sheet) Find the following probabilities: 1. P(z < 1) 2. P(z < 0)

3. P(z < 1.5)

4. P(z > 1.5)

5. P(2.3 < z < 3.1)

Why do we use z-scores? To answer questions such as… 1. Suppose teachers at LHS have an age distribution X ~ N(40, 8). What is the likelihood that a randomly selected teacher from this population would have an age of 24 or younger?

Why do we use z-scores? 2. Assuming the same distribution exists for age of teachers at LHS, how likely is it for a randomly selected teacher from LHS to be older than 50 years of age?

Why do we use z-scores? 3. Again using the same teacher age distribution at LHS, what is the probability that a randomly selected teacher’s age would fall somewhere between 34 and 50 years of age?

Why do we use z-scores? 4. Again using the same teacher age distribution at LHS, at what age do 25% of the teachers fall below?

AP Statistics Tuesday, September 5, 2011 • Standard Normal Distribution (calculator) • Turn on / Link “Catalog Help” 2nd VARS (DIST) 2:normalcdf( 3:invNorm( • HW: Worksheet

(gives you a list of distributions we will use) lower bound, upper bound [, µ, α] area [, µ, α]

Area to the LEFT

Z-Scores on the CALCULATOR 1. Suppose teachers at LHS have an age distribution X ~ N(40, 8). What is the likelihood that a randomly selected teacher from this population would have an age of 24 or younger?

Z-Scores on the CALCULATOR 2. Assuming the same distribution exists for age of teachers at LHS, how likely is it for a randomly selected teacher from LHS to be older than 50 years of age?

Z-Scores on the CALCULATOR 3. Again using the same teacher age distribution at LHS, what is the probability that a randomly selected teacher’s age would fall somewhere between 34 and 50 years of age?

Z-Scores on the CALCULATOR 4. Again using the same teacher age distribution at LHS, at what age do 25% of the teachers fall below?

Z-Scores on the CALCULATOR 5. Again using the same teacher age distribution at LHS, at what age do 10% of the teachers fall above?

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