MATH 221 WEEK 6 LIVE LECTURE

B Heard

(Don’t copy or post without my permission, students may download a copy for personal

use)

MATH 221 WEEK 6 LIVE LECTURE The following are examples of the Week

6 Lab, please note that I CHANGED the data!

MATH 221 WEEK 6 LIVE LECTURE1. When rolling a die, is this an example

of a discrete or continuous random variable? Explain your reasoning.

You should be able to answer this Read about discrete and continuous variables

MATH 221 WEEK 6 LIVE LECTURE2. Calculate the mean and standard

deviation of the probability distribution created by rolling a four sided die. Either show work or explain how your answer was calculated.

For the sake of example, I am going to use a four-sided die (your lab deals with a six-sided die)

MATH 221 WEEK 6 LIVE LECTURE In other words, my die would look like a

pyramid. You could roll a 1,2,3 or 4 and they are all equally likely.

MATH 221 WEEK 6 LIVE LECTURE To get the mean

Mean = ƩxP(x) = 1(1/4)+2(1/4)+3(1/4)+4(1/4) = 10/4 or 2.5

Die Value times the probabilityThere are four equally likely sides, so the probability for each would be ¼.

MATH 221 WEEK 6 LIVE LECTURE To get the standard deviation

St. Dev. = sqrt((1-2.5)2(1/4)+ (2-2.5)2(1/4)+ (3-2.5)2(1/4)+ (4-2.5)2(1/4) = sqrt(1.250) = 1.118

REMEMBER WE WERE DEALING WITH A FOUR SIDED DIE

MATH 221 WEEK 6 LIVE LECTURE3. Give the mean for the mean column of

the Worksheet. Is this estimate centered about the parameter of interest (the parameter of interest is the answer for the mean in question 2)?

I CHANGED THE DATA

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE Type the word “Mean” to the right of Die10

MATH 221 WEEK 6 LIVE LECTURE Pull up Calc > Row Statistics and select

the radio-button corresponding to Mean. For Input variables: enter all 10 rows of the die data. Go to the Store result in: and select the Mean column. Click OK and the mean for each observation will show up in the Worksheet.

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE Now I have the means calculated in the

“Mean” column

MATH 221 WEEK 6 LIVE LECTURE We also want to calculate the median

for the 10 rolls of the die. Label the next column in the Worksheet with the word Median. Repeat the above steps but select the radio-button that corresponds to Median and in the Store results in: text area, place the median column.

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE Same process as Mean, except choose

Median radial button and change Store result to Median by double clicking on Median in your list on the left.

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE Calculate descriptive statistics for the

mean and median columns that where created above. Pull up Stat > Basic Statistics > Display Descriptive Statistics and set Variables: to mean and median. The output will show up in your Session Window. Print this information.

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURESo number 3 wants the “Mean of Means” (From previous chart)Mean of means = 2.64 yes, this is

generally centered around the parameter of interest (the 2.5 I calculated in number 2)

Honestly I would have liked for it to be a tad closer (but remember I changed data at a whim and probably put too many 3’s and 4’s in rather than actually rolling a 4-sided die)

MATH 221 WEEK 6 LIVE LECTURE4. Give the mean for the median column

of the Worksheet. Is this estimate centered about the parameter of interest (the parameter of interest is the answer for the mean in question 2)?

 Mean of medians = 2.775, this is

definitely farther away from the parameter of interest (the 2.5 I calculated mathematically in number 2)

MATH 221 WEEK 6 LIVE LECTURE5. Give the standard deviation for the mean

and median column. Compare these and be sure to identify which has the least variability?

 StDev of means = 0.3202StDev of medians = 0.472The standard deviation of the means

is smaller, thus it has less variability than the medians. This would mean the data for the means is grouped closer together.

MATH 221 WEEK 6 LIVE LECTURE6. Based on questions 3, 4, and 5 is the

mean or median a better estimate for the parameter of interest? Explain your reasoning.

 In my case, the mean seems to be a

better estimate because it is closer to the mathematically calculated mean and the standard deviation is less than that of the medians meaning the means are grouped closer together.

MATH 221 WEEK 6 LIVE LECTURE7. Give and interpret the 95% confidence

interval for the hours of sleep a student gets.

  I changed the data! So these are not

the answers to your lab!

MATH 221 WEEK 6 LIVE LECTURE We are interested in calculating a 95%

confidence interval for the hours of sleep a student gets. Pull up Stat > Basic Statistics > 1-Sample t and set Samples in columns: to Sleep. Click the OK button and the results will appear in your Session Window.

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE

MATH 221 WEEK 6 LIVE LECTURE

Therefore, the 95% confidence interval would be (6.232, 8.168). I would be 95% confident that the true mean number of hours a student sleeps would be between those two values.

I changed the data! So these are not the answers to your lab!

MATH 221 WEEK 6 LIVE LECTURE8. Give and interpret the 99% confidence

interval for the hours of sleep a student gets.

(Same approach as number 7, but MAKE SURE you click options and change confidence to 99%)

MATH 221 WEEK 6 LIVE LECTURE9. Compare the 95% and 99% confidence

intervals for the hours of sleep a student gets. Explain the difference between these intervals and why this difference occurs.

 The 99% confidence interval is wider

than the 95%, which is always the case. To get more confidence, the bounds widened (i.e. it’s the only way you can get more certainty).