14.310x: Data Analysis for Social Scientists Special Distributions, the Sample Mean, and the Central Limit Theorem

Welcome to your sixth homework assignment! You will have about one week to work through the assignment. We encourage you to get an early start, particularly if you still feel you need more experience using R. We have provided this PDF copy of the assignment so that you can print and work through the assignment offline. You can also go online directly to complete the assignment. If you choose to work on the assignment using this PDF, please go back to the online platform to submit your answers based on the output produced. Some of the questions we are asking are not easily solvable using math so we recommend you to use your R knowledge and the content of previous homework assignments to find numeric solutions.

Good luck!

Suppose that 𝑋! 𝑖. 𝑖. 𝑑. 𝑈[0, 𝜃]. You want to build a 90% confidence interval for 𝜃. To do so, you will need an estimator for 𝜃 and you will need to know the estimator’s distribution. Let’s consider 𝜃, = "#$

" 𝑋("). (Remember that 𝑋(") is the nth order statistic). This estimator is a

variant on the MLE. We have used the 𝑛$% order statistic, which is the MLE, but multiplied it by "#$ "

to remove its bias. Its PDF is "!"#

("#$)! (!$#

)! for 𝑥 ∈ [0, "#$

" 𝜃] and 0 otherwise.

Question 1 Let a be a function of 𝑛 and 𝜃 such that 5% of the distribution of 𝜃% is to the left of a. a is then given by

𝑎 = √𝐴! 𝐵 𝐶 𝜃

What is the value of A? o 𝜃 o 𝑛 o 𝑛 + 1 o 0.05

What is the value of B? o 𝜃 o 𝑛 o 𝑛 + 1 o 0.05

What is the value of C? o 𝜃 o 𝑛 o 𝑛 + 1 o 0.05

Question 2

Let b be a function of 𝑛 and 𝜃 such that 5% of the distribution of 𝜃% is to the right of b. b is then given by

𝑏 = √𝐴! 𝐵 𝐶 𝜃

What is the value of A? o 𝜃 o 𝑛 o 𝑛 + 1 o 0.95

What is the value of B? o 𝜃 o 𝑛 o 𝑛 + 1 o 0.95

What is the value of C? o 𝜃 o 𝑛 o 𝑛 + 1 o 0.95

Question 3 Using those values that you found above, what is a probability statement of the form 𝑃-𝑎 < 𝜃% < 𝑏1 = .90 as a function of 𝑛 and 𝜃?

o 𝑃 7√0.95 ! "#$" 𝜃 ≤ 𝜃 , ≤ √0.05 ! "#$" 𝜃 ; = .90

o 𝑃 7√0.95 ! ""#$𝜃 ≤ 𝜃 , ≤ √0.05 ! "#$" 𝜃 ; = .90

o 𝑃 7√0.05 ! ""#$𝜃 ≤ 𝜃 , ≤ √0.95 ! "#$" 𝜃 ; = .90

o 𝑃 7√0.05 ! "#$" 𝜃 ≤ 𝜃 , ≤ √0.95 ! "#$" 𝜃 ; = .90

Question 4 If you rearrange the quantities in the probability statement so that 𝜃 is alone in the middle, bracketed by functions of the random sample and known quantities, what would be this probability statement?

o 𝑃 7 ) *

√,../ ! " "#$

≤ 𝜃 ≤ ) *

√,.,/ ! "#$ "#"

; = .90

o 𝑃 7 ) *

√,../ ! " "#$

≤ 𝜃 ≤ ) *

√,.,/ ! " "#$

; = .90

o 𝑃 7 ) *

√,../ ! " "#$

≤ 𝜃 ≤ ) *

√,.,/ ! "#$ " ; = .90

o 𝑃 7 ) *

√,.,/ ! " "#$

≤ 𝜃 ≤ ) *

√,../ ! " "#$

; = .90 We are going to show this in R. We have provided you with this code that demonstrates this. Question 5 In the code, there are three symbols standing in for specific values: XXX, YYY, and ZZZ. Which of the following values correspond to XXX, YYY, and ZZZ?

o 𝑋𝑋𝑋 = 𝜃, 𝑌𝑌𝑌 = 1, 𝑍𝑍𝑍 = & o 𝑋𝑋𝑋 = 𝜃, 𝑌𝑌𝑌 = 1, 𝑍𝑍𝑍 = | o 𝑋𝑋𝑋 = 𝜃, 𝑌𝑌𝑌 = 2, 𝑍𝑍𝑍 = & o 𝑋𝑋𝑋 = 𝑛, 𝑌𝑌𝑌 = 2, 𝑍𝑍𝑍 = & o 𝑋𝑋𝑋 = 𝑛, 𝑌𝑌𝑌 = 2, 𝑍𝑍𝑍 = |

We invite you to run this code to see that it is true that in 90% of the simulated samples this confidence interval (CI) contains the real value of 𝜃. You can play with the code, changing both the value of 𝜃 and the sample size. Now suppose 𝑋& 𝑖. 𝑖. 𝑑. 𝑁(𝜇, 4) and 𝑛 = 25. We want to test 𝐻': 𝜇 = 0 𝑣𝑠. 𝐻(: 𝜇 ≠ 0. Question 6 Given the information above, what test statistic should you use?

o The minimum o The sample variance o The maximum o The sample mean

Question 7 Choose the figure below whose shaded region corresponds to the critical region.

o

o

o

o Question 8 Compute 𝛼 (the probability that we reject the null hypothesis when the null is true) as a function of the critical value(s). Note: A critical value is the boundary between the critical region and the rest of the sample space. In the example in class, we denoted the critical value k.

𝛼 = F1 − 𝛷 I 2𝑘 5 KL

𝛼 = 2F1 − 𝛷 I 5𝑘 2 KL

𝛼 = F1 − 𝛷 I 5𝑘 2 KL

𝛼 = 2F1 − 𝛷 I 2𝑘 5 KL

Question 9

Is there a connection between conducting a hypothesis test for 𝜇 = 𝜇' at a significance level 𝛼 and finding a (1 − 𝛼) confidence interval for a parameter 𝜇? Yes No

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A B C D E F G H I J K L M N O P Q

House Type

Modern $9,505,000

Condo $13,205,000

Ranch $18,515,000

Mediterranean $15,855,000

Modern 17%

Condo 23%

Ranch 32%

Mediterranean 28%

Sales by House Type

Modern Condo Ranch Mediterranean

$0

$2,000,000

$4,000,000

$6,000,000

$8,000,000

$10,000,000

$12,000,000

$14,000,000

$16,000,000

$18,000,000

$20,000,000

Modern Condo Ranch Mediterranean

Sales by House Type

32% of total

Summary by Type

SOLUTION SAMPLE

$0

$100,000

$200,000

$300,000

$400,000

$500,000

$600,000

3/21/2016 5/10/2016 6/29/2016 8/18/2016 10/7/2016 11/26/2016 1/15/2017 3/6/2017 4/25/2017

1 and 2 Bedroom Condo Sales

SOLUTION SAMPLE

Date of Purchase House Type Purchase Price Agent Buyers BedroomsBathrooms Fireplace Near SchoolsRate Bank Mortgage Years

3/4/2017 Condo $425,000 Remax Wilson 2 2 Yes No 5.75% People's 15

3/4/2017 Condo $385,000 By Owner Zimmerman 2 1 Yes No 6.75% Bank Of America 30

9/17/2016 Condo $225,000 Remax Jackson 2 2 No Yes 5.50% First Federal 30

9/17/2016 Condo $355,000 Century 21 Grande 2 2 No Yes 6.50% First Federal 30

9/15/2016 Condo $440,000 By Owner Chipman 1 2 No Yes 6.75% Citywide 30

7/16/2016 Condo $550,000 Century 21 Holland 2 1 Yes No 5.50% Citizens 30

7/16/2016 Condo $550,000 Remax Nettle 2 1 No Yes 6.25% First Federal 40

7/16/2016 Condo $550,000 By Owner Hawkins 2 1 Yes No 6.75% People's 15

4/20/2016 Condo $385,000 Homes and Garden Melon 2 1 No Yes 5.75% Nations 15

4/20/2016 Condo $425,000 Homes and Garden Perry 1 1 Yes No 6.75% Bank Of America 30

Total $4,290,000 10 1.8 1.4 6.23% 26.5SOLUTION SAMPLE

A Skills Approach: Excel 2016 Chapter 7: Exploring Advanced Charts and Graphics

1 | Page Challenge Yourself 7.4 Last Updated 3/2/18

Challenge Yourself 7.4 In this project, you will analyze housing trends using charting tools.

Skills needed to complete this project: • Changing the Sparkline Type

• Adding Markers to Sparklines

• Applying Quick Styles and Other Formatting to Sparklines

• Formatting Other Chart Elements

• Emphasizing a Data Point in a Pie Chart

• Formatting a Data Point or a Data Series

• Inserting a Shape

• Applying Quick Styles and Other Formatting to Shapes

• Adding and Removing Trendlines

• Changing Trendline Options

• Inserting a Picture

• Applying Quick Styles and Other Formatting to Pictures

• Resizing and Moving Pictures

• Adding a Picture to a Header

This image appears when a project instruction has changed to accommodate an update to

Microsoft Office 365. If the instruction does not match your version of Office, try using the alternate instruction

instead.

Important: Download the resource file(s) needed for this project from the Resources link. Make sure to

extract the file(s) after downloading the resources zipped folder. Visit the SIMnet instant help for step-by-

step instructions.

1. Open the start file EX2016-ChallengeYourself-7-4. The file will be renamed automatically to

include your name. Change the project file name if directed to do so by your instructor, and save it

2. If the workbook opens in Protected View, click the Enable Editing button in the Message Bar at the

top of the workbook so you can modify the workbook.

3. On the Summary by Type worksheet, modify the Sparklines.

a. Change the Sparklines to columns.

b. Apply the Quick Style Green, Sparkline Style Accent 6 (no dark or light) (the first style at the

right in the third row of the gallery).

4. Add a marker to the Sparklines.

a. Add a marker for the high point.

b. Change the high point marker color to Blue (the third color from the right in the row of standard

colors).

Step 1

Download start file

A Skills Approach: Excel 2016 Chapter 7: Exploring Advanced Charts and Graphics

2 | Page Challenge Yourself 7.4 Last Updated 3/2/18

5. In the pie chart, change the colors for the individual data labels to match their respective data points

and then apply a bevel shape effect to all the data labels. Selecting a single data label is similar to

selecting a single data point.

a. Apply the Blue, Accent 1, Lighter 80% fill color (the fifth color from the left in the second row

of theme colors) to the data label for the Modern data point.

b. Apply the Orange, Accent 2, Lighter 80% fill color (the fifth color from the right in the second

row of theme colors) to the data label for the Condo data point.

c. Apply the White, Background 1, Darker 5% fill color (the first color at the left in the second

row of theme colors) to the data label for the Ranch data point.

d. Apply the Gold, Accent 4, Lighter 80% fill color (the third color from the right in the second

row of theme colors) to the data label for the Mediterranean data point.

e. Select all the data labels and apply the Round or Circle bevel shape effect (the first effect in the

Bevel section).

6. Emphasize the Ranch data point.

a. Rotate the pie chart 240° until the Ranch data point is at the right side.

b. Explode the Ranch data point away from the pie by 35%.

c. Apply the Round or Circle bevel shape effect (the first effect in the Bevel section) to the Ranch

data point. Hint: Do not use the Format Data Point task pane to apply the shape effect. Use the

option off the Ribbon instead.

7. In the column chart, change the color of each data point to match the corresponding color in the pie

chart. You do not need to change the color of the Modern data point.

a. In the column chart, apply the Orange, Accent 2 fill color (the fifth color from the right in the

top row of theme colors) to the Condo data point.

b. In the column chart, apply the White, Background 1, Darker 35% fill color (the first color at

the left in the fifth row of theme colors) to the Ranch data point.

c. In the column chart, apply the Gold, Accent 4 fill color (the third color from the right in the first

row of theme colors) to the Mediterranean data point.

8. Add a callout to the column chart to emphasize the Ranch data point.

a. Add a Callout: Line shape (the fifth option in the Callouts section of the Gallery) to the chart.

b. Add the text: 32% of total

c. Resize and position the callout so it is pointing to the Ranch column as shown in Figure EX 7.66.

A Skills Approach: Excel 2016 Chapter 7: Exploring Advanced Charts and Graphics

3 | Page Challenge Yourself 7.4 Last Updated 3/2/18

FIGURE EX 7.66

d. Change the shape style Subtle Effect - Gray, Accent 3 (the fourth style from the left in the

fourth row of theme styles).

9. Apply the Fill: Blue, Accent color 1; Shadow WordArt style (the second WordArt style) to the

column chart title.

Apply the Fill - Blue, Accent 1, Shadow WordArt style (the second WordArt style)

to the column chart title.

10. Apply the Dash dashes line option (the fourth option in the list) to the column chart gridlines.

11. Add a trendline to the 1 and 2 Bedroom Condo Sales chart on the Condo Sales worksheet.

a. Switch to the Condo Sales worksheet.

b. Add a trendline.

c. Change the trendline type to a two period moving average trendline.

12. Add the SOLD picture from your local drive, network, or external media storage device, and then

apply a style, resize and move the picture.

a. Insert the SOLD picture downloaded from the Resources link.

b. Apply the Metal Frame picture Quick Style (the third style from the left in the top row of the

style gallery) to the picture.

c. Resize the picture so it is 1.5 inches wide. Do not change the aspect ratio of the picture.

d. Reposition the picture so it is located near the bottom right corner of the chart as shown in

Figure EX 7.67.

13. Add a picture from your local drive, network, or external media storage device to the worksheet

header.

a. Add the RealtyLogo picture (downloaded from the Resources link) to the left section of the

worksheet header. Hint: Because this worksheet is a chart sheet, you cannot switch to Page

Layout view. You will need to use the Page Setup dialog instead. If you need help with this

method, refer to the another method box in the Adding a Picture to a Header skill.

Download

Resources

Download

Resources

A Skills Approach: Excel 2016 Chapter 7: Exploring Advanced Charts and Graphics

4 | Page Challenge Yourself 7.4 Last Updated 3/2/18

b. Preview how the worksheet will look when printed to verify that you inserted the correct picture

into the header. It should look similar to Figure EX 7.67.

FIGURE EX 7.67

14. Save and close the workbook.

15. Upload and save your project file

16. Submit project for grading.

Step 2

Upload & Save

Step 3

Grade my Project

Summary by Type

Sales by House Type

Modern Condo Ranch Mediterranean 9505000 13205000 18515000 15855000 Modern Condo Ranch Mediterranean 9505000 13205000 18515000 15855000

Sales by House Type

Modern Condo Ranch Mediterranean 9505000 13205000 18515000 15855000

Condo Sales

1 and 2 Bedroom Condo Sales

Purchase Price 42798 42798 42630 42630 42628 42567 42567 42567 42480 42480 425000 385000 225000 355000 440000 550000 550000 550000 385000 425000

Sales Data