Programming challenges

PC2

You may need to edit these first lines to work on your own machine. Note that for working with .Rmd files interactively in Rstudio you may find it easier to do this using the drop down menus: “Session” → “Set Working Directory” → “To Source File Location”

## setwd("~/Documents/Teaching/2019/stats/")
## list.files("data/week_04")

mobile <- read.csv("data/week_04/COS-Statistics-Mobile_Sessions.csv")
total <- read.csv("data/week_04/COS-Statistics-Gov-Domains-Only.csv")

I’ll write a little function to help inspect the data. Make sure you understand what the last line of the function is doing.

summary.df <- function (d) {
    print(nrow(d))
    print(ncol(d))
    print(head(d))
    print(d[sample(seq(1, nrow(d)), 5),])
}

Then I can run these two lines a few times to look at some samples

summary.df(mobile)
## [1] 231
## [1] 8
##   Operating_System Sessions New_Sessions New_Users Bounce_Rate
## 1              iOS   332291        47.75    158674       60.79
## 2          Android   170107        45.53     77453       58.14
## 3          Windows    27325        44.76     12231       44.60
## 4    Windows Phone    10109        45.71      4621       59.01
## 5       BlackBerry     1375        39.27       540       62.98
## 6        (not set)      408        83.09       339       72.30
##   PagesPerSession AvgSessionDuration                  Month
## 1            2.34            0:02:11 01/01/2015 12:00:00 AM
## 2            2.98            0:03:53 01/01/2015 12:00:00 AM
## 3            3.26            0:02:40 01/01/2015 12:00:00 AM
## 4            2.14            0:01:45 01/01/2015 12:00:00 AM
## 5            2.10            0:02:24 01/01/2015 12:00:00 AM
## 6            1.82            0:01:01 01/01/2015 12:00:00 AM
##     Operating_System Sessions New_Sessions New_Users Bounce_Rate
## 182 Playstation Vita        6       100.00         6       100.0
## 214          Android   214077        47.17    100978        59.4
## 194          Android   178625        47.32     84526        57.4
## 92          Series40        8       100.00         8       100.0
## 14  Playstation Vita        6       100.00         6       100.0
##     PagesPerSession AvgSessionDuration                  Month
## 182            1.00            0:00:00 01/01/2016 12:00:00 AM
## 214            3.65            0:04:45 07/01/2016 12:00:00 AM
## 194            3.65            0:04:48 03/01/2016 12:00:00 AM
## 92             1.00            0:00:00 07/01/2015 12:00:00 AM
## 14             1.00            0:00:00 01/01/2015 12:00:00 AM
summary.df(total)
## [1] 1242
## [1] 7
##                    domain pageviews unique.pageviews average.time.on.page
## 1        www.seattle.gov/   3525737          2689843              0:01:19
## 2       www2.seattle.gov/   2158182           125984              0:01:12
## 3       web6.seattle.gov/    367871           204803              0:01:18
## 4 spdblotter.seattle.gov/    117645            91076              0:01:14
## 5       web1.seattle.gov/     79529            32258              0:01:09
## 6       find.seattle.gov/     78611            62516              0:00:39
##   bounce.rate exit.percent                  month
## 1       50.86        36.53 04/01/2015 12:00:00 AM
## 2       41.69         4.53 04/01/2015 12:00:00 AM
## 3       40.66        23.23 04/01/2015 12:00:00 AM
## 4       69.29        46.42 04/01/2015 12:00:00 AM
## 5       59.57        18.76 04/01/2015 12:00:00 AM
## 6       25.67        21.74 04/01/2015 12:00:00 AM
##                         domain pageviews unique.pageviews
## 542 dpdwinw101.ad.seattle.gov/        52               25
## 678        murray.seattle.gov/     41246            35629
## 776   consultants.seattle.gov/      2790             2203
## 808  perspectives.seattle.gov/        46               44
## 644                                   NA               NA
##     average.time.on.page bounce.rate exit.percent                  month
## 542               125.76         0.0        11.54 07/01/2015 12:00:00 AM
## 678              0:02:38         0.8         0.69 09/01/2015 12:00:00 AM
## 776              0:01:10      5446.0      3391.00 10/01/2015 12:00:00 AM
## 808              0:02:17      8667.0      4130.00 10/01/2015 12:00:00 AM
## 644                               NA           NA

I can check for missing values and summarize the different columns using lapply:

lapply(total, summary)
## $domain
##                                               
##                                            34 
##                             2035.seattle.gov/ 
##                                            15 
##                          artbeat.seattle.gov/ 
##                                            15 
##                    atyourservice.seattle.gov/ 
##                                            15 
##                          bagshaw.seattle.gov/ 
##                                            15 
##                       bottomline.seattle.gov/ 
##                                            15 
##                       brainstorm.seattle.gov/ 
##                                            15 
##              buildingconnections.seattle.gov/ 
##                                            15 
##                  centerspotlight.seattle.gov/ 
##                                            15 
##                        cityclerk.seattle.gov/ 
##                                            15 
##                            clark.seattle.gov/ 
##                                            15 
##                            clerk.seattle.gov/ 
##                                            15 
##                    climatechange.seattle.gov/ 
##                                            15 
##                           conlin.seattle.gov/ 
##                                            15 
##                      consultants.seattle.gov/ 
##                                            15 
##                          council.seattle.gov/ 
##                                            15 
##                             find.seattle.gov/ 
##                                            15 
##                         fireline.seattle.gov/ 
##                                            15 
##                       frontporch.seattle.gov/ 
##                                            15 
##                           godden.seattle.gov/ 
##                                            15 
##                 grantsandfunding.seattle.gov/ 
##                                            15 
##                       greenspace.seattle.gov/ 
##                                            15 
##                   hackthecommute.seattle.gov/ 
##                                            15 
##                   humaninterests.seattle.gov/ 
##                                            15 
##                           licata.seattle.gov/ 
##                                            15 
##                          married.seattle.gov/ 
##                                            15 
##                      mayormcginn.seattle.gov/ 
##                                            15 
##                                m.seattle.gov/ 
##                                            15 
##                             news.seattle.gov/ 
##                                            15 
##                           obrien.seattle.gov/ 
##                                            15 
##                        onthemove.seattle.gov/ 
##                                            15 
##                         parkways.seattle.gov/ 
##                                            15 
##                     perspectives.seattle.gov/ 
##                                            15 
##                       powerlines.seattle.gov/ 
##                                            15 
##                        rasmussen.seattle.gov/ 
##                                            15 
##                          rectech.seattle.gov/ 
##                                            15 
##                           sawant.seattle.gov/ 
##                                            15 
##                         sdotblog.seattle.gov/ 
##                                            15 
##                  sdotperformance.seattle.gov/ 
##                                            15 
##                       seattlerdy.seattle.gov/ 
##                                            15 
##                       spdblotter.seattle.gov/ 
##                                            15 
##                         techtalk.seattle.gov/ 
##                                            15 
##                       thebuyline.seattle.gov/ 
##                                            15 
##                         thescoop.seattle.gov/ 
##                                            15 
##                             web6.seattle.gov/ 
##                                            15 
##                             www2.seattle.gov/ 
##                                            15 
##                        www.clerk.seattle.gov/ 
##                                            15 
##                            wwwqa.seattle.gov/ 
##                                            15 
##                           cmstrn.seattle.gov/ 
##                                            14 
##                             cms8.seattle.gov/ 
##                                            13 
##                           igxqa8.seattle.gov/ 
##                                            13 
##                                  seattle.gov/ 
##                                            13 
##                            cttab.seattle.gov/ 
##                                            12 
##                          okamoto.seattle.gov/ 
##                                            12 
##                             web5.seattle.gov/ 
##                                            12 
##                             web7.seattle.gov/ 
##                                            12 
##                        education.seattle.gov/ 
##                                            11 
##                             web1.seattle.gov/ 
##                                            11 
##                           webqa7.seattle.gov/ 
##                                            11 
##                             www4.seattle.gov/ 
##                                            11 
##                            alert.seattle.gov/ 
##                                            10 
##                           alerts.seattle.gov/ 
##                                            10 
##                             data.seattle.gov/ 
##                                            10 
##             seattle-govstat.demo.socrata.com/ 
##                                            10 
##                          connect.seattle.gov/ 
##                                             9 
##                             igx8.seattle.gov/ 
##                                             9 
##                           murray.seattle.gov/ 
##                                             9 
##                           webqa6.seattle.gov/ 
##                                             9 
##                              www.seattle.gov/ 
##                                             9 
##           www.seattle.gov.googleweblight.com/ 
##                                             9 
##                          alphaqa.seattle.gov/ 
##                                             8 
##                          cmsdev8.seattle.gov/ 
##                                             8 
##                    dpdwinw101.ad.seattle.gov/ 
##                                             8 
##          web6.seattle.gov.googleweblight.com/ 
##                                             8 
##                              cms.seattle.gov/ 
##                                             7 
##                             ctab.seattle.gov/ 
##                                             7 
##                     www.citylink.seattle.gov/ 
##                                             7 
##                   aboveandbeyond.seattle.gov/ 
##                                             6 
##                         citylink.seattle.gov/ 
##                                             6 
##                        langstoninstitute.org/ 
##                                             6 
##                      mayormurray.seattle.gov/ 
##                                             6 
##                    take21.seattlechannel.org/ 
##                                             6 
##                             web8.seattle.gov/ 
##                                             6 
##                           wwwdev.seattle.gov/ 
##                                             6 
##                        www.evergreenapps.org/ 
##                                             6 
##                     www.safeyouthseattle.org/ 
##                                             6 
##                            cityofseattle.gov/ 
##                                             5 
##                councilconnection.seattle.gov/ 
##                                             5 
##                     filmandmusic.seattle.gov/ 
##                                             5 
##                         gonzalez.seattle.gov/ 
##                                             5 
##                         homebase.seattle.gov/ 
##                                             5 
##                          igxdev8.seattle.gov/ 
##                                             5 
##                        www.mayor.seattle.gov/ 
##                                             5 
## www.seattle.gov.offcampus.lib.washington.edu/ 
##                                             5 
##                  capitalprojects.seattle.gov/ 
##                                             4 
##                    dpdwina307.ad.seattle.gov/ 
##                                             4 
##                          herbold.seattle.gov/ 
##                                             4 
##                          johnson.seattle.gov/ 
##                                             4 
##                           juarez.seattle.gov/ 
##                                             4 
##                                       (Other) 
##                                            97 
## 
## $pageviews
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##       1      24     402   66417    2752 4172985      34 
## 
## $unique.pageviews
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##       1      17     285   28515    2204 3213093      34 
## 
## $average.time.on.page
## 0:00:00         0:01:11 0:01:18    0.00 0:01:12 0:01:13 0:01:14 0:01:20 
##     134      34      17      17      16      15      13      13      12 
## 0:01:53 0:01:09 0:01:17 0:01:23 0:01:32 0:01:05 0:01:24 0:01:29 0:01:36 
##      12      11      11      11      11      10      10      10      10 
## 0:01:51 0:01:54 0:01:58 0:00:55 0:01:01 0:01:06 0:01:08 0:01:10 0:01:16 
##      10      10      10       9       9       9       9       9       9 
## 0:01:22 0:01:25 0:01:30 0:01:35 0:01:37 0:01:56 0:00:39 0:00:53 0:00:56 
##       9       9       9       9       9       9       8       8       8 
## 0:00:57 0:01:03 0:01:27 0:01:31 0:01:38 0:01:43 0:01:47 0:00:42 0:00:48 
##       8       8       8       8       8       8       8       7       7 
## 0:01:07 0:01:19 0:01:40 0:01:41 0:01:42 0:01:45 0:01:50 0:01:52 0:02:00 
##       7       7       7       7       7       7       7       7       7 
## 0:02:04 0:02:31 0:00:31 0:00:54 0:00:59 0:01:21 0:01:26 0:01:44 0:01:48 
##       7       7       6       6       6       6       6       6       6 
## 0:01:59 0:02:06 0:02:07 0:02:23 0:02:35 0:00:08 0:00:38 0:01:00 0:01:02 
##       6       6       6       6       6       5       5       5       5 
## 0:01:04 0:01:33 0:01:34 0:01:39 0:01:46 0:02:09 0:02:12 0:02:19 0:02:21 
##       5       5       5       5       5       5       5       5       5 
## 0:02:27 0:02:29 0:02:42 0:02:47 0:02:51 0:02:54 0:03:03 0:00:11 0:00:12 
##       5       5       5       5       5       5       5       4       4 
## 0:00:20 0:00:27 0:00:33 0:00:41 0:00:49 0:00:50 0:00:58 0:01:15 0:01:28 
##       4       4       4       4       4       4       4       4       4 
## (Other) 
##     350 
## 
## $bounce.rate
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.     NA's 
##     0.00    24.89    65.75   430.47    79.32 10000.00       34 
## 
## $exit.percent
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.     NA's 
##     0.00    17.67    42.09   347.91    62.37 10000.00       34 
## 
## $month
##                        01/01/2015 12:00:00 AM 01/01/2016 12:00:00 AM 
##                     34                     84                     84 
## 02/01/2015 12:00:00 AM 02/01/2016 12:00:00 AM 03/01/2015 12:00:00 AM 
##                     78                     79                     80 
## 03/01/2016 12:00:00 AM 04/01/2015 12:00:00 AM 04/01/2016 12:00:00 AM 
##                     88                     83                     87 
## 05/01/2015 12:00:00 AM 06/01/2015 12:00:00 AM 07/01/2015 12:00:00 AM 
##                     75                     84                     85 
## 08/01/2015 12:00:00 AM 09/01/2015 12:00:00 AM 10/01/2015 12:00:00 AM 
##                     70                     84                     77 
## 12/01/2015 12:00:00 AM 
##                     70
lapply(mobile, summary)
## $Operating_System
##                           Android             Bada       BlackBerry 
##               34               17                4               17 
##       Firefox OS              iOS               LG              LGE 
##                5               10               12                1 
##              MOT     Nintendo 3DS            Nokia        (not set) 
##                1                7               16               17 
## Playstation Vita          Samsung         Series40        SymbianOS 
##               12               17               10               17 
##          Windows    Windows Phone 
##               17               17 
## 
## $Sessions
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##       6      16     217   38469   10718  519563      34 
## 
## $New_Sessions
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    0.44   45.53   84.62   72.65  100.00  100.00      34 
## 
## $New_Users
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##       6      13     124   17575    4853  236550      34 
## 
## $Bounce_Rate
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    0.00   53.85   62.98   66.21   84.62  100.00      34 
## 
## $PagesPerSession
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   1.000   1.210   1.860   2.082   2.500   9.000      34 
## 
## $AvgSessionDuration
## 0:00:00         0:00:06 0:00:41 0:01:06 0:01:21 0:01:45 0:01:50 0:02:00 
##      46      34       3       3       3       3       3       3       3 
## 0:00:04 0:00:09 0:00:25 0:00:42 0:01:01 0:01:05 0:01:07 0:01:09 0:01:20 
##       2       2       2       2       2       2       2       2       2 
## 0:01:46 0:01:56 0:02:02 0:02:06 0:02:40 0:02:49 0:03:01 0:03:05 0:03:53 
##       2       2       2       2       2       2       2       2       2 
## 0:00:02 0:00:14 0:00:17 0:00:20 0:00:21 0:00:24 0:00:26 0:00:29 0:00:32 
##       1       1       1       1       1       1       1       1       1 
## 0:00:34 0:00:38 0:00:40 0:00:43 0:00:44 0:00:46 0:00:48 0:00:49 0:00:50 
##       1       1       1       1       1       1       1       1       1 
## 0:00:52 0:00:55 0:00:56 0:01:03 0:01:08 0:01:12 0:01:14 0:01:16 0:01:19 
##       1       1       1       1       1       1       1       1       1 
## 0:01:24 0:01:25 0:01:26 0:01:28 0:01:29 0:01:33 0:01:34 0:01:35 0:01:37 
##       1       1       1       1       1       1       1       1       1 
## 0:01:41 0:01:42 0:01:51 0:01:52 0:01:54 0:02:01 0:02:03 0:02:05 0:02:08 
##       1       1       1       1       1       1       1       1       1 
## 0:02:09 0:02:10 0:02:11 0:02:13 0:02:14 0:02:15 0:02:17 0:02:18 0:02:19 
##       1       1       1       1       1       1       1       1       1 
## 0:02:24 0:02:26 0:02:34 0:02:39 0:02:47 0:02:48 0:02:52 0:02:56 0:02:57 
##       1       1       1       1       1       1       1       1       1 
## 0:03:04 0:03:07 0:03:14 0:03:18 0:03:21 0:03:25 0:03:26 0:03:29 0:03:36 
##       1       1       1       1       1       1       1       1       1 
## (Other) 
##      22 
## 
## $Month
##                        01/01/2015 12:00:00 AM 01/01/2016 12:00:00 AM 
##                     34                     15                      9 
## 02/01/2015 12:00:00 AM 02/01/2016 12:00:00 AM 03/01/2015 12:00:00 AM 
##                     13                     11                     15 
## 03/01/2016 12:00:00 AM 04/01/2015 12:00:00 AM 04/01/2016 12:00:00 AM 
##                      9                     12                     10 
## 05/01/2015 12:00:00 AM 06/01/2015 12:00:00 AM 07/01/2015 12:00:00 AM 
##                     11                     14                     12 
## 07/01/2016 12:00:00 AM 08/01/2015 12:00:00 AM 08/01/2016 12:00:00 AM 
##                      9                     14                     10 
## 09/01/2015 12:00:00 AM 10/01/2015 12:00:00 AM 12/01/2015 12:00:00 AM 
##                     10                     12                     11

PC3

First let’s create a table/array using tapply that sums pageviews per month across all the sites:

total.views.bymonth.tbl <- tapply(total$pageviews, total$month, sum)
total.views.bymonth.tbl
##                        01/01/2015 12:00:00 AM 01/01/2016 12:00:00 AM 
##                     NA                6350440                3471121 
## 02/01/2015 12:00:00 AM 02/01/2016 12:00:00 AM 03/01/2015 12:00:00 AM 
##                5820453                3366834                6609602 
## 03/01/2016 12:00:00 AM 04/01/2015 12:00:00 AM 04/01/2016 12:00:00 AM 
##                4087054                6481483                3644750 
## 05/01/2015 12:00:00 AM 06/01/2015 12:00:00 AM 07/01/2015 12:00:00 AM 
##                6544055                6952488                8084318 
## 08/01/2015 12:00:00 AM 09/01/2015 12:00:00 AM 10/01/2015 12:00:00 AM 
##                7045189                3067760                2961681 
## 12/01/2015 12:00:00 AM 
##                5745045

If you run class on total.views.bymonth.tbl you’ll notice it’s not a data frame yet. We can change that:

total.views <- data.frame(months=names(total.views.bymonth.tbl),
                          total=total.views.bymonth.tbl)

head(total.views)
##                                        months   total
##                                                    NA
## 01/01/2015 12:00:00 AM 01/01/2015 12:00:00 AM 6350440
## 01/01/2016 12:00:00 AM 01/01/2016 12:00:00 AM 3471121
## 02/01/2015 12:00:00 AM 02/01/2015 12:00:00 AM 5820453
## 02/01/2016 12:00:00 AM 02/01/2016 12:00:00 AM 3366834
## 03/01/2015 12:00:00 AM 03/01/2015 12:00:00 AM 6609602

Let’s cleanup the rownames (this would all work the same if i didn’t do this part).

rownames(total.views) <- NULL

head(total.views)
##                   months   total
## 1                             NA
## 2 01/01/2015 12:00:00 AM 6350440
## 3 01/01/2016 12:00:00 AM 3471121
## 4 02/01/2015 12:00:00 AM 5820453
## 5 02/01/2016 12:00:00 AM 3366834
## 6 03/01/2015 12:00:00 AM 6609602

PC4

Onwards to the mobile dataset!

Here we have a challenge because we have to estimate total pageviews (it’s not given in the raw dataset). I’ll do this by multiplying sessions by pages-per-session. This assumes that the original pages-per-session calculation is precise, but I’m not sure what else we could do under the circumstances.

mobile$total.pages <- mobile$Sessions * mobile$PagesPerSession 

Then, making the views-per-month array is more or less copy/pasted from above:

mobile.views.bymonth.tbl <- tapply(mobile$total.pages, mobile$Month, sum)
mobile.views.bymonth.tbl
##                        01/01/2015 12:00:00 AM 01/01/2016 12:00:00 AM 
##                     NA              1399185.6               668275.2 
## 02/01/2015 12:00:00 AM 02/01/2016 12:00:00 AM 03/01/2015 12:00:00 AM 
##              1275315.2               592607.8              1402086.4 
## 03/01/2016 12:00:00 AM 04/01/2015 12:00:00 AM 04/01/2016 12:00:00 AM 
##               800842.8              1381295.1               788533.7 
## 05/01/2015 12:00:00 AM 06/01/2015 12:00:00 AM 07/01/2015 12:00:00 AM 
##              1605914.9              1722519.5              1988848.0 
## 07/01/2016 12:00:00 AM 08/01/2015 12:00:00 AM 08/01/2016 12:00:00 AM 
##               878142.6              1741067.8               912435.4 
## 09/01/2015 12:00:00 AM 10/01/2015 12:00:00 AM 12/01/2015 12:00:00 AM 
##               564453.5              1285288.0              1223414.0
mobile.views <- data.frame(months=names(mobile.views.bymonth.tbl),
                           mobile=mobile.views.bymonth.tbl)
rownames(mobile.views) <- NULL

PC5

Now we merge the two datasets. Notice that I have created the months column in both datasets with exactly the same name.

views <- merge(mobile.views, total.views, all.x=TRUE, all.y=TRUE, by="months")

These are sorted in strange ways and will be difficult to graph because the dates are stored as characters. Let’s convert them into Date objects. Then I can use sort.list to sort everything.

views$months <- as.Date(views$months, format="%m/%d/%Y %H:%M:%S")

views <- views[sort.list(views$months),]

Take a look at the data. Some rows are missing observations. We can drop those rows using complete.cases:

lapply(views, summary)
## $months
##         Min.      1st Qu.       Median         Mean      3rd Qu. 
## "2015-01-01" "2015-05-01" "2015-09-01" "2015-09-20" "2016-02-01" 
##         Max.         NA's 
## "2016-08-01"          "1" 
## 
## $mobile
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##  564454  800843 1275315 1190013 1402086 1988848       1 
## 
## $total
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
## 2961681 3557936 5820453 5348818 6576828 8084318       3
views[rowSums(is.na(views)) > 0,]
##        months   mobile total
## 13 2016-07-01 878142.6    NA
## 15 2016-08-01 912435.4    NA
## 1        <NA>       NA    NA
views.complete <- views[complete.cases(views),]

PC6

For my proportion measure, I’ll take the mobile views divided by the total views.

views.complete$prop.mobile <- views.complete$mobile / views.complete$total

PC7.

library(ggplot2)
ggplot(data=views.complete) + aes(x=months, y=prop.mobile) + geom_point() + geom_line() + scale_y_continuous(limits=c(0, 1))

  1. For my estimate of the proportion I’ll just calculate an average from the monthly numbers:
mean(views.complete$prop.mobile)
## [1] 0.2308486
  1. From the graph, this proportion seems quite stable with the exception of a single outlier month in late 2015.

Statistical questions

SQ1 — 4.8

The general formula for a confidence interval is \(point~estimate~±~z^*\times~SE\). First, identify the three different values. The point estimate is 45%, \(z^* = 2.58\) for a 99% confidence level (that’s the number of standard deviations around the mean that ensure that 99% of a Z-score distribution is included), and \(SE = 2.4\%\).

With this we can plug and chug:

\[52\% ± 2.58 \times 2.4\% → (45.8\%, 58.2\%)\]

From this data we are 99% confident that between 45.8% and 58.2% U.S. adult Twitter users get some news through the site.

SQ2 — 4.10

  1. False. See the answer to 4.8 above. With \(\alpha = 0.01\), we can consult the 99% confidence interval. It includes 50% but also goes lower.

  2. False. The standard error of the sample does not contain any information about the proportion of the population included in the sample. It measures the variability of the sample distribution.

  3. False. Increasing the sample size will decrease the standard error. Consider the formula: \(\frac{\sigma}{\sqrt{n}}\). A smaller \(n\) will result in a larger standard error.

  4. False. All else being equal, a lower confidence interval will cover a narrower range. A higher interval will cover a wider range. To confirm this, revisit the formula in SQ1 above. and plug in the corresponding alpha value of .9, resulting in a \(z^*\) value of 1.28 (see the Z-score table in the back of OpenIntro).

SQ3 — 4.19

The hypotheses should be about the population mean (\(\mu\)) and not the sample mean (\(\bar{x}\)). The null hypothesis should have an equal sign. The alternative hypothesis should be about the critical value, not the sample mean. The following would have been better:

\[H_0: \mu = 10~hours\] \[H_A: \mu \gt 10~hours\]

SQ4 — 4.32

  1. True. See part (d) of SQ2 above.
  2. False. A lower alpha value is the probability of Type 1 Error, so reducing the one reduces the other.
  3. False. Failure to reject the null is evidence that we cannot conclude that the true value is different from the null. This is very different from evidence that the null hypothesis is true.
  4. True. Consult the section of OpenIntro discussing statistical power and Type 2 Error.
  5. True. We’ll revisit this in a moment below, but consider the relationship between statistical test, the standard error, and the sample size. As the sample size increases towards infinity, the standard error approaches zero, resulting in arbitrarily precise point estimates that will result in rejecting the null hypothesis for any value of a test statistic for any critical value of \(\alpha\).

Empirical paper questions

EQ1

In my words (or rather formulas since I think that’s less ambiguous), the key pairs of null/alternative hypotheses look something like the following:

Let \(\Delta\) be the parameter estimate for the difference in mean percentage of positive (\(\mu_{pos}\)) and negative (\(\mu_{neg}\)) words between the experimental and control conditions for the treatments of reduced negative content (\(R_{neg}\) and reduced positive content (\(R_{pos}\)).

For the reduced negative content conditions (the left-hand side of Figure 1), the paper tests:

\[HR_{neg}1_0: \Delta_{\mu_{pos}} = 0\] \[HR_{neg}1_a: \Delta{\mu_{pos}} \gt 0\] And: \[HR_{neg}2_0: \Delta_{\mu_{neg}} = 0\] \[HR_{neg}2_a: \Delta_{\mu_{neg}} \lt 0\] Then, for the reduced positive content conditions (the right-hand side of Figure 1), the paper tests:

\[HR_{pos}1_0:~~ \Delta_{\mu_{pos}} = 0\] \[HR_{pos}1_a:~~ \Delta{\mu_{pos}} \lt 0\]

And:

\[HR_{pos}2_0:~~ \Delta_{\mu_{neg}} = 0\] \[HR_{pos}2_a:~~ \Delta_{\mu_{neg}} \gt 0\] Note that the theories the authors used to motivate the study imply directions for the alternative hypotheses, but nothing in the description of the analysis suggests that they used one-tailed tests. I’ve written these all in terms of specific directions here to correspond with the theories stated in the paper. They could also (arguably more accurately) have been written in terms of inequalities (“\(\neq\)”).

EQ2

The authors’ estimates suggest that reduced negative News Feed content causes an increase in the percentage of positive words and a decrease in the percentage of negative words in subsequent News Feed posts by study participants (supporting \(HR_{neg}1_a\) and \(HR_{neg}2_a\) respectively).

They also find that reduced positive News Feed content causes a decrease in the percentage of negative words and an increase in the percentage of positive words in susbequent News Feed posts (supporting \(HR_{pos}1_a\) and \(HR_{pos}2_a\))

EQ3

Cohen’s \(d\) puts estimates of experimental effects in standardized units (much like a Z-score!) in order to help understand their size relative to the underlying distribution of the dependent variable(s). The d-values for each of the effects estimated in the paper are 0.02, 0.001, 0.02, and 0.008 respectively (in the order presented in the paper, not in order of the hypotheses above!). These are miniscule effects. However, the treatment itself is also quite narrow in scope, suggesting that the presence of any treatment effect at all is an indication of the underlying phenomenon (emotional contagion). Personally, I find it difficult to attribute much substantive significance to the results because I’m not even convinced that tiny shifts in the percentage of positive/negative words used in News Feed updates accurately index meaningful emotional shifts (maybe we could call it linguistic contagion instead?). Despite these concerns and the ethical considerations that attracted so much public attention, I consider this a clever, well-executed study and I think it’s quite compelling. I expect many of you will have different opinions of various kinds!