Jan 21 Statistic for the day:The width of train tracks is

4 feet 8.5 inches.Why?

Assignment: Read Chapter 9Assignment: Read Chapter 9

Exercises from Chapter 8: 16, 18Exercises from Chapter 8: 16, 18

These slides were created by Tom Hettmansperger and in some cases modified by David Hunter

Research Question 1: How high should I build my doorways so that 99% of the people will not

have to duck?

Secondary Question 2: If I built my Secondary Question 2: If I built my doors 75 inches (6 feet 3 inches) doors 75 inches (6 feet 3 inches) high, what percent of the people high, what percent of the people

would have to duck?would have to duck?

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Histogram of Height, with Normal Curve

Question 1Question 2

Find the value at Question 1 so that 99% of the distributionis below it.The value at Question 2 is 75; find the amount of distributionabove it.

Z-Scores: Measurement in Standard Deviations

Given the mean (68), the standard deviation Given the mean (68), the standard deviation (4), and a value (height say 75) compute (4), and a value (height say 75) compute

This says that 75 is 1.75 standard deviations above the mean.

Z = (75-mean) / SD = (75-68) / 4 = 1.75

Answer to Question 2: What percent of people would have to duck if I built my doors 75 inches high?

Recall: 75 has a Z-score of 1.75

From the standard normal table in the book: .96 or 96% of the distribution is below 1.75. Hence, .04 or 4% is above 1.75.

So 4% of the distribution is above 75 inches.

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Histogram of Height, with Normal Curve

Question 2

4% in here

75

96% in here

(GIVEN)

(FIND IN TABLE)

The value at Question 2 is 75; find the amount of distributionabove it. Convert 75 to Z = 1.75 and use Table 8.1 in book.

Question 1: What is the value so that 99% of thedistribution is below it? Called the 99th percentile.

1. Look up the Z-score that corresponds to the 99th percentile. From the table: Z = 2.33.

2. Now convert it over to inches:

Since 77 inches is 6 feet 5 inches, 99% of the distributionis shorter than 77 inches and they will not have to duck.

2.33 = (h – 68)/4

h = 68 + 2.33(4) = 77.3

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Histogram of Height, with Normal Curve

Question 177.3 inches is the 99th percentile

99% in here

(GIVEN)

(FIND):

Find the value at Question 1 so that 99% of the distributionis below it. Look up Z-score for 99th percentile and convert it back to inches.

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Compare Heights of Females and Males

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Heights in Inches ( red circle is my doorway 77 inches)

78

Shaquille O’Neal is 7 feet 1 inch or 85 inches tall. How many people in the country are taller?

1. We will assume that heights are normally distributedwith mean 68 inches and standard deviation 4 inches.

2. O’Neal’s Z-score is Z = (85-68)/4 = 4.25. In other wordsO’Neal is 4.25 standard deviations above the mean! We would generally consider him from a different population.

3. There is .000011 above 4.25 standard deviations.

3. There are roughly 250 million people in US.

48.8% are over the age of 20.

That is 122 million.

4. Hence, there should be roughly

.000011 times 122 million

or 1342 people taller than Shaquille O’Neal

What is your Z-Score?

What is your percentile?

How many PSU students are taller?

Who Is the Tallest Person in Class?

Suppose someone claims to have tossed a coin 100 times and got 70 heads. Would you believe them? We need to know what the distribution of We need to know what the distribution of

the number of heads in 100 tosses looks the number of heads in 100 tosses looks like for a fair coin.like for a fair coin.

We need the mean and standard deviation We need the mean and standard deviation for this distribution.for this distribution.

1. What is the mean?

2. What is the standard deviation?

3. Let’s suppose the smooth version is bell shaped.

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Toss a coin 100 timesRepeat 500 times and form a histogram

Number of heads

So the distribution of the number of heads in 100 tosses of a fair coin is:

Roughly normal, mean about 50, SD about 5Roughly normal, mean about 50, SD about 5 What is the Z-score of 70?What is the Z-score of 70? Ans: 4Ans: 4 What is the percentile?What is the percentile? Ans: .999968 or 99.9968%Ans: .999968 or 99.9968% Now do you believe them?Now do you believe them? NONO Weighted coin is a BETTER explanationWeighted coin is a BETTER explanation