Elements of Statistics (Math 106) - Exam 1Fall 2005 - Brad Hartlaub

Name

Directions: Please answer all of the questions below and show your work. The point values for eachproblem are indicated in parentheses. You may use one sheet of formulas and any software that isavailable on the Kenyon network during the exam. Good luck and have a nice break!

1. One of the difficulties faced by courts of law is to get those registered voters who receive asummons to actually appear at the courthouse for jury duty. The percentage of individuals thatreport for jury duty may change over time both within years and across years. In the FranklinCounty Municipal Court in Columbus, Ohio, jury duty is two weeks long. Thus, a new group ofpotential jurors must be brought in twenty-six times a year. The percentage of individuals thatreported for jury duty for nine years, 1985 and 1997 through 2004, are provided in the filep:\data\math\stats\FCMC Percentages.mtw. The reporting dates vary slightly from year-to-year sothey are coded sequentially from 1, the first group to report in January, to 26, the last group toreport in December. A variety of methods have been used over the years to increase participationrates.

L

IA

0 V0 -ill- 0N 0

N-..,."00 -co-

M0N 0

-.. NN00

--[C]-N

N * 00

-.. NPI00N -[] 0

-.. 0

0 N

00 0N -IT]-- 0

-.. 0en NenenPI

-oJ--..COenenPI

OJ00

-.. *.......enenPI

r-....-..

fl" 1'1an L_ ... ..... __ .1coenPI"-

* -m-Lt')000

..,0- I I I I I I I I I I

C- o 0 0 0 0 0 0 0 0 0>< 0 00 r-.... \D Lt') V M N

0m Iqea

...

Descriptive Statistics: 1985, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3

1985 26 0 22.135 0.890 4.537 14.200 20.000 21.700 24.800

1997 26 0 48.10 3.52 17.94 23.30 34.50 41.55 60.60

1998 26 0 67.40 1.55 7.90 50.70 62.45 65.16 70.78

1999 26 0 65.47 1.01 5.16 55.50 61.98 65.70 69.13

2000 26 0 85.24 1.82 9.27 63.70 79.30 87.30 93.04

2001 25 1 84.92 2.13 10.64 68.10 74.50 88.70 94.10

2002 26 0 85.73 2.06 10.50 50.00 81.60 90.00 92.80

2003 25 1 89.06 1.40 7.00 76.30 83.20 89.10 96.45

2004 26 0 86.761 0.936 4.772 75.830 82.383 87.375 90.858

Variable Maximum1985 33.0001997 88.401998 83.601999 75.202000 97.002001 100.002002 98.402003 100.002004 94.640

Descriptive statistics are provided below for the reporting percentages from 1998 and 2000.

Descriptive Statistics: Reporting Date, 1998, 2000

VariableReporting Date19982000

Mean13.5067.4085.24

StDev7.657.909.27

Correlations: Reporting Date, 1998

Pearson correlation of Reporting Date and 1998 = -0.647

Correlations: Reporting Date, 2000

Pearson correlation of Reporting Date and 2000 = -0.632

(b) Use only the descriptive statistics above to compute the least squares regression line forpredicting the reporting percentage from the coded reporting date in 1998. Show your work.(10) J r~ 7..90 - ) - (- ,)J d

b:: J :: -.6'17 j(,{ t: -. ~l,g/ ~ = y... b)C .:: {'7tj- -.~(,9;(11. ~s~ ~ 7t.4/g-1 ~ Iy" rerr/.."J fJt'Ce./.-r: 7b.m -. u, f (~rrf.7~

(c) Use Minitab to compute the least squares regression line for predicting the reporting percentagefrom the coded reporting date in 2000. (5)

/ 9: 9f.~ -. 7{,{(~cr/1p~)~I~~}-~ (d) Interpret the value of the slope for your estimated model in 2000. (5) ~ J //

<:~t! s~t+15tiO'.Jerp) ~r A 11AV/1'; (z uveel:.) ('4creo.<;(? 1'1 repor/"l cI~ ~e

c}J:~ P- S~~'fJ t:pe. I().rlrCtpcj,(}Vfrde deCrfWes '1. 76~%, .

::t~s.ffe) Identifythe predictedvalueandresidualforthe fourthreportingdate in 2000.(

(5)

re~/-44/= ~ rf<j()J>-; 97- 92. fo 723,

(t) Identify the percentage of variation in the response variable that is explained by using theexplanatory variable. (5)

2. The Mental Development Index (MDI) of the Bayley Scales of Infant Development is astandardized measure used in observing infants over time. It is approximately normal with a mean

of 100 and a standard deviation of 16 tI~~(6)

(a) What proportion of children has_MDI of at most 90?I

'l6 (CD

(b) What proportion of children has MDI of at least 120? (5)

~ /-,97'1150 =-,/05650(DO 12L>

(c) Find the MDI score such that only 1% of the population has an MDI sCj)fe below it. (5)

,e' ~ ~- - l,-1vcdF~ 62. ??r,/(d) What linear transformation will change MDI into a new variable xnew= a + bMDI that has a

mean of75 and a standard deviation of 16? (10)

3. Did you have your morning coffee? A study on the effect of caffeine on memory involved havingsubjects take a memory test 20 minutes after drinking a cup of coffee. Subjects were randomlyassigned to drink decaffeinated coffee, regular coffee (with caffeine), or a "half-caP' mixture of thetwo (getting a half dose of caffeine). For each subject, a test score (the number of items recalledcorrectly) was recorded. The subjects were not told which type of coffee they had been given, butthe researchers for the study prepared the cups of coffee themselves (out of sight of the subjects).

(a) Identify the treatment and response variables and the levels of the treatment in r.Hisstudy. (5)

~alAe-l ~ 4.MCUv/ o/' <:Ci/4~~ Leve4 0/' h~(!~"

4prp VOl,..,atle= let! {'(Oft' tl! doref/! 0:Li4,,~ h'af1~(b) Place a check mark next to the descriptors that apply to this study (2 points each) /l() (tCLl,4f'"ft.

o Observational" Randomized

o ReplicatedII) Controlledo Double-blind

(c) If the researchers wished to account for differences between women and nlen in their responseto caffeine, should they use a matched-pairs or blocked design? (5)

4. The number of undergraduates at Kenyon is approximately 2,500, while the number at Ohio Stateis approximately 40,000. A simple random sample of 125 students in each undergraduateclass(freshman, sophomore, junior, senior) at each institution is obtained to estimate the proportion,p,of all students who feel that smoking should be banned from all public areas on campus.

l

(d) ~f::~ ~~u~~:~~~~Z~~~~~~~I~~h~fs~~;:~~~:~sO~~i~:c~~::~ at eac~inst~, ratherthana

~~~;ameSam~l~g variability, compared to the sample ProPorti~~~~ma~~~t~e;sE~~~~ut

I;: wf' CU~tA""'P / If 4/lroy/"""a.4f 4 {'a<#' .,/l)i~j( I~~/,/~V]~ --de f'4"'Yl, vqna'ifMil

be (a7er -tr -Me {d""fk IrejJO~b>1 1/19.01 #/0>1

be('(tIA(P ./1tf' I?VR.,-al! fO"'!," ('IEp aI d1'''''

(. oJ) )( 2~OO =: 12~) ",;,;1 be AIde.' {<44~'- -tK." ~fam/le {'i~e a/ tJ/,;; ~ (. ;~c<JO)JpnJer $'a..,If file .;::> );r5 vanaG, #d {";';k,.. sa<1(/~{r?f ""I'JfOf"f','/1/ Va.1f~'h~I

5. Many statistical methods implicitly assume a normal data model, but real data often exhibitdepartures from normality. The left column presents four data sets: the sizes of Arctic charr (aspecies of fish) from a lake in Scotland, the population density of nations around the world, thelength of songs by the band Led Zeppelin, and the body masses of mammal species around theworld. Based on the shape of the distribution, draw lines to match each histogram to its normalprobability plot on the right. (10 - 2.5 for each).

Size of matur@ArcticcharTcolle(;ted Inloch Tay, Scotland

25 .. . ...20

Populationdensity of nations00

70 2-,

'"

20

10

100 200 300 ..oJ 9)Q 6IXIMoan population density 19.50 - 2000 (per 5qU8rakin)

Led Zeppelin song lengths

20 '.

15 .. ~...

6 8lrack tength (minutes) -3

Bodymasses of mammalspecies.0

30

10

1.2 2.4 3.6 9Body mass (90logarithmic seafe)

6.0-3