Math - Statistics Test

/4 1). Suppose a 12-year-old asked you, “I heard someone use some words I don’t understand. Can you tell me what the difference is between a statistic and a parameter?”

a). Write an answer (using either a few sentences or bullet points) to their question.

b). After thanking you for your help, they follow up with another question, “Now that you’ve described them, I think I’ve heard of a statistic before, but I don’t think I’ve heard people talk about parameters. Why don’t people use parameters more often?” Write an answer to this question as you did in part a.

/3 2). A drug company is testing a new aspirin and finds it succeeds in relieving all migraine headache symptoms in twelve hours. Using statistical methods, it is shown an aspirin with no effect has a likelihood of getting these results in roughly 35 chances in 100.

a). Do the results appear to have statistical significance? Why or why not?

b). Do the results appear to have practical significance? Why or why not?

/4 3). You are put in charge of creating a survey for the DC Department of Motor Vehicles focusing on aggressive driving. Without writing actual questions, give one example each of data you could collect which would be classified as: a). Categorical.

b). Quantitative; discrete.

c). Quantitative; continuous.

/2 4). In a study case done at a university, researchers tested 2,500 African American women every five years for 30 years for diabetes. It was found 14.7% of the women developed diabetes over the course of the testing. Was this an example of an observational study or an experiment? Justify your answer.

/3 5). For the two examples given below, identify the type of observational study (retrospective, cross-sectional, prospective). Justify each of your answers.

a). Pregnant women who were survivors of the World Trade Center terrorist attack of September 11th, 2001 were observed, as were their children, through the year 2011 to see if mother, child or both developed signs of Post-Traumatic Stress Disorder.

b). The Nielsen Media Research Company will survey 5,000 households on 9/30/2017 to determine the proportion of households that watch Saturday Night Live.

/5 6). Questions included on a survey for incoming freshmen are given below. For each question, identify what level of measurement (nominal, ordinal, interval or ratio) would be most appropriate for each piece of requested data. Justify each of your answers. a). In what state is your home address located?

b). What is your current class standing (Freshman, Sophomore, Junior, Senior)?

c). What was the price of your most expensive textbook this semester?

/3 7). Students in the dormitories of a small, private university in the state of New York live in suites of four double rooms. There are 48 suites, with eight students per suite.

a). The suites are numbered 1 – 48, twelve of the suites are chosen and all eight residents of the suites were interviewed. What sampling technique is demonstrated here? Justify your answer.

b). Interviewers stand outside the cafeteria and talk to the first 96 students who enter. What sampling technique is demonstrated here? Justify your answer.

/6 8). Two histograms are given below showing the ages of the Oscar winners for Best Actress/Actor through 2008. On the left is the histogram for actresses, actors are on the right. Use the histograms to answer the questions below.

a). How many actresses and how many actors are in this observational study?

b). What type of distribution (normal, uniform, skewed left or skewed right) would you say best fits the actresses distribution on the histogram to the left? Justify your answer.

c). What type of distribution (normal, uniform, skewed left or skewed right) would you say best fits the actors distribution on the histogram to the right? Justify your answer.

d). Does there seem to be a difference in the age when actresses and actors win this Oscar? Justify your answer.

/7 9). At a local retailer, television sales were totaled and recorded for each month. Below is a list of television sales by month.

Find the following information, showing all work. Round according to the rounding rule for measures of central tendency and remember to include units.

a). 𝑥̅

b). 𝑥̃

c). mode(s) if any

d). midrange

e). Which measure of central tendency would be best to use for this data set? Justify your answer.

/9 10). A bank is working on improving customer service and is attempting to address customer wait time in particular. On Day 1, the bank has customers stay in one single line and go up to the next available teller. On Day 2, the bank has customers get in individual lines for each teller and stay in that lane until they have been served. There were an equal number of tellers on both days. Employees measured wait time (in minutes) on both days and the following is a sample of six customers from each day.

a). Find the coefficient of variation for Day 1 showing all work. Round to one decimal.

b). Find the coefficient of variation for Day 2 showing all work. Round to one decimal.

c). If the bank wants to give their customers the most consistent experience, which method would you recommend they use? Justify your answer.

/6 11). For all of the MATH 110 classes taught by Professor Kraft, Test #1 had the following measures: 𝜇 = 77.62, 𝜎 = 11.27. Assuming the grades were normally distributed, answer the following showing all work:

a). What percentage of grades would be between 55.08 and 100.16?

b). 68% of the grades in the class would be between what two grades?

c). Why does the problem use 𝜇 and 𝜎 instead of 𝑥̅ and 𝑠?

/6 12). Scores on the SAT test have a mean of 1518 and a standard deviation of 325. Scores on the ACT test have a mean of 21.1 and a standard deviation of 4.8. Janine takes both exams and receives a grade of 2290 on her SAT and 30 on her ACT.

a). While proud of both scores, which test did Janine perform better on? Justify your answer and show all work.

b). Are either of Janine’s scores unusual? Justify your answer.

/6 13). Answer the questions below using the boxplot.

Ages of Oscar Winners for “Best Actress”

30

40

50

60

20

70

80

Age (in Years)

a). What are the ages for the middle 50% of “Best Actress” Oscar winners?

b). 75% of “Best Actress” Oscar winning actresses win by what age?

c). What type of distribution (normal, uniform, skewed left or skewed right) does this data have? Justify your answer.

d). Does this data appear to have any outliers? If so, are they high or low end outliers? Justify your answer.

e). What is the range of ages for “Best Actress” Oscar winners? Show work.

Formulas

Standard Deviation:

̅ ∑(𝑥 − 𝜇)2

𝑠 𝜎 =

√

𝑁

OR

𝑠

Coefficient of Variation (round to one decimal place):

𝑠 𝜎

𝐶𝑉 = ∙ 100% 𝐶𝑉 = ∙ 100%

𝑥̅ 𝜇

𝒛 Score / Standard Score / Standard Value (round to two decimal places):

𝑥 − 𝑥̅ 𝑥 − 𝜇

𝑧 = 𝑧 =

𝑠 𝜎

Percentile of a value:

number of values less than 𝑥

Percentile of value 𝑥 = ∙ 100

total number of values

Location of a Percentile in a Data Set:

𝑘

𝐿 = () 𝑛

100

Outliers:

Low Outlier < 𝑄1 − 1.5 × 𝐼𝑄𝑅

High Outlier > 𝑄3 + 1.5 × 𝐼𝑄𝑅