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lab_4_probabilities.doc

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Lab 4 – Probabilities

Use Statcrunch to help you answer the following questions about our Statistics class. Note that it will NOT

be able to answer the questions for you, but it can help, especially if you sort the data. To sort data, click

on Data/Sort, then choose all columns, then under Sort criteria select the column you want to sort on. Do

NOT try to sort on two different columns at the same time because the rest of the data will no longer match

up. Instead, look at the two columns in the data table to answer the question.

1. What proportion of students in this class are shorter than 66 inches?

answer?

What proportion are taller than 66 inches?

How did you find the

What proportion are exactly 66

inches tall?

2. What is the z-score for someone in this class who is 66 inches tall?

How did you find the

answer?

3. According to the z-table, in what percentile is a person who is 66 inches tall?

4. Do your answers to #1 and #3 match?

Why do you think they do/don’t match?

Thinking back to Labs 1 and 2 where you looked at a histogram of our class heights, would you say

that our class is normally distributed?

5. What proportion of students in this class have a GPA above 3.5?

6. What is the probability that a randomly selected student in this class has a GPA above 3.5?

7. What is the probability that a randomly selected student in this class is computer savvy? (i.e. Rated

themselves as 7 or higher)

8. What is the probability that a randomly selected student in this class has a GPA above 3.5 and is

computer savvy?

How did you find this answer?

Note: You cannot simply multiply

your answers to questions 6 and 7!

9. Is the probability that a student in this class has a GPA above 3.5 and is computer savvy (your

answer to question #8) equal to the probability that a student in this class has a GPA above 3.5

(your answer to question #6) times the probability that a student in this class is computer savvy?

(your answer to question #7)

Why or why not?

10. What is the probability that a randomly selected student in this class does not smoke?

11. What is the probability that a randomly selected student in this class has brown eyes?

12. What is the probability that a randomly selected student in this class does not smoke or has brown

eyes?

13. Is the probability that a student in this class does not smoke or has brown eyes equal to the

probability that a student does not smoke plus the probability that a student has brown eyes minus

the probability that a student does not smoke and has brown eyes?

14. What is the probability that a randomly selected student in this class is female?

15. What is the probability that a randomly selected student in this class is a female basketball fan?

16. What is the probability that a randomly selected student in this class is a basketball fan given that

she is female?

17. Given that a student is female, what is the probability that she has blue eyes?

18. Given that a student has blue eyes, what is the probability that she is female?

19. What is the probability that a randomly selected student in this class is shorter than you?

20. What is the probability that a randomly selected student in this class is shorter than you given that

she is female?

Bonus Question: What is the probability that a randomly selected sample of 3 students from this class

are all shorter than you?

Explain how you got your answer:

Lab 3 – Association

Part I – Categorical Variables

Have Statcrunch give you a Contingency Table for Eye Color and Favorite Sport from our class data by

clicking on Stats/Tables/Contingency/With Data. After you’ve selected the variables, you will see a

number of checkboxes on Statcrunch– choose the first four checkboxes – Row percent, Column percent,

Percent of total, and Expected count – so that they appear in your report. Finally, click on “Calculate” and

use the table Statcrunch produces to answer the following questions:

Between eye color and favorite sport:

1. What percent of blue-eyed students like basketball?

Note: You are not being asked what

percent of all students like basketball, so you will use either row percent or column percent in

answering the question, depending on the order in which you chose the variables. Which

percent did you use?

2. How many students like basketball?

How many brown-eyed students like basketball?

How many brown-eyed students are expected to like basketball?

How many

basketball fans have brown eyes?

Are the expected counts for basketball the same for all

eye colors?

Are the expected counts for eye color the same for all sports?

3. What percent of brown-eyed students like baseball?

What percent of baseball fans have

brown eyes?

What percent of the entire class likes baseball?

What percent of the

entire class has brown eyes?

4. Does there appear to be an association between eye color and favorite sport? Give a reason for

your answer based on the numbers in the contingency table.

Keep your contingency table open and have Statcrunch create a side-by-side bar graph for you.

After clicking on Graphs/Bar Plot/With Data, select either favorite sport or eye color from the

list, and then Group By the other variable. (Note: Since there are fewer eye colors than sports,

grouping by favorite sport will produce fewer charts.) At the bottom of the window under “For

Multiple Graphs” you can tell Statcrunch how many rows or columns per page you want. There

should be 5 different eye colors, so type in 5 to get them all on one page. Finally, click on Create

Graph, and then compare the bar graphs to the numbers in your contingency table.

Recall from the lesson video that we really can’t determine if there is an association between two

categorical variables unless there are at least 5 observations in each cell of the contingency table.

Do we meet this condition for eye color and favorite sport?

Can you tell from the bar graph if there is an association between the variables?

Why or

why not?

Because there are not enough students in our class to be able to draw any valid conclusions about the

association between any two categorical variables, we will now look at an example that will enable us to do

so:

An unusually severe increase in gasoline prices may have motivated full-sized pickup truck

buyers to purchase a highly fuel-efficient vehicle. Purchase behavior was collected in one area

for one year and reported below.

Low Fuel Prices

High Fuel Prices

Number of Highly

Fuel Efficient Trucks

Purchased

392

442

Number of Ordinary

Cars and Regular

Trucks Purchased

36,929

42,255

Total

Total

a. Complete the row and column totals in the table above. What is the independent

variable? Explain why.

b. Create a relative frequency table below by typing in the proportion of vehicles in each

category using the column numbers from the table above.

Low Fuel Prices

High Fuel Prices

Proportion of

Highly Fuel Efficient

Trucks Purchased

Proportion of

Ordinary Cars and

Regular Trucks

Purchased

c. Is there an association between fuel prices and the number of highly fuel-efficient trucks

purchased?

Give a reason for your answer based on the results in the table.

Now have Statcrunch give you a Contingency Table for Gender and Exercise using the same process

you used above for Eye Color and Favorite Sport. Complete the numbers table below based on the

data provided in the Contingency Table:

Males

Number who

exercise regularly

or everyday

Number who

exercise rarely or

never

Females

Total

Total

Now convert the numbers table into a percentage table as you did for the fuel price vs type of vehicle

example:

Males

Females

Proportion who

exercise regularly

or everyday

Proportion who

exercise rarely or

never

Is there an association between gender and exercise? Be sure to give a reason for your answer based

on the percentages in the table above.

Part II – Quantitative Variables

Now have Statcrunch run a regression analysis on the following pairs of quantitative variables. Click on

Stat/Regression/Simple Linear. Since we are only looking at correlation in this section, it doesn’t really

matter for now which variable is the explanatory (X) and which is the response (Y). Click on Calculate,

then use the results to complete the following chart:

Here’s a helpful hint to save you time when completing this chart: Find the correlation coefficient for the

whole class and put it in the chart, then click on the Options button in the top left corner of the Regression

Results window and choose EDIT. You will be taken back one step to a screen where you can now Group

By Gender. You can now get the correlation coefficient for each gender to complete that line in the chart.

Between the Variables:

Coefficient of Correlation

Whole class

by Gender

M

F

Height and Shoe Size

GPA and Fastest You’ve Ever Driven

GPA and Computer Savvy

Using your answers in the chart above, does there appear to be a correlation between:

Height and Shoe Size? Give a reason based on the correlation coefficient.

GPA and Fastest You’ve Ever Driven? Give a reason.

GPA and Computer Savvy? Give a reason.

Part III – Correlation and You

Now let’s see how you correlate with the rest of the class:

1. Redo the regression for height and shoe size making height the explanatory variable (x) and shoe size

the response variable (y).

Copy the regression equation which Statcrunch gives you here:

2. What is the slope of this regression line?

(Hint: Recall from high school Algebra that, given an

equation in the form y=mx+b, the slope is the number in front of x. The slope is important because it tells

us how the response variable changes for each unit change in the explanatory variable.)

3. Now plug your own height into the regression equation and calculate (by hand) the shoe size this

regression line predicts for you. (Show your work!):

4. Now take your actual shoe size and subtract the predicted value which you just calculated:

This last number represents your RESIDUAL for shoe size using the regression equation.

5. Now find your residual for Fastest You’ve Ever Driven based on GPA. Show your work by completing

the following steps:

1. Get the regression equation from Statcrunch. Make GPA the explanatory variable (x) and Fastest

Driven the response variable (y):

2. Calculate your predicted driving speed by plugging your GPA into the above equation. (Again,

show your work!):

3. Find your residual (show the two numbers you subtracted to get it):

…

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