1.1 Definitions of Statistics, Probability, and Key Terms
Use the following information to answer the next five exercises. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new drug is currently under study to address a respiratory virus. It is given to patients once the patient exhibits symptoms of the virus. Of interest is the average (mean) length of time in days from the time the patient starts the treatment until the symptoms are alleviated. Two researchers each follow a different set of 40 patients with the respiratory virus from the start of treatment until the symptoms are alleviated. The following data (in days) are collected.
Researcher A:3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34
Researcher B:3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29
Determine what the key terms refer to in the example for Researcher A.
sample
statistic
1.2 Data, Sampling, and Variation in Data and Sampling
“Number of times per week” is what type of data?
a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous
Use the following information to answer the next four exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Antonio, Texas. The first house in the neighborhood around the park was selected randomly, and then the resident of every eighth house in the neighborhood around the park was interviewed.
“Duration (amount of time)” is what type of data?
a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous
The colors of the houses around the park are what kind of data?
a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous
The population is ______________________
Table 1.27 contains the total number of deaths worldwide as a result of earthquakes over a 13-year period.
| Year | Total Number of Deaths |
|---|---|
| 1 | 231 |
| 2 | 21,357 |
| 3 | 11,685 |
| 4 | 33,819 |
| 5 | 228,802 |
| 6 | 88,003 |
| 7 | 6,605 |
| 8 | 712 |
| 9 | 88,011 |
| 10 | 1,790 |
| 11 | 320,120 |
| 12 | 21,953 |
| 13 | 768 |
| Total | 823,856 |
Use Table 1.27 to answer the following questions.
- What is the proportion of deaths between Year 8 and Year 13?
- What percent of deaths occurred before Year 2?
- What is the percent of deaths that occurred in Year 4 or after Year 11?
- What is the fraction of deaths that happened before Year 13?
- What kind of data is the number of deaths?
- Earthquakes are quantified according to the amount of energy they produce (examples are 2.1, 5.0, 6.7). What type of data is that?
- What contributed to the large number of deaths in Year 11? In Year 5? Explain.
For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
A group of test subjects is divided into twelve groups; then four of the groups are chosen at random.
The first 50 people who walk into a sporting event are polled on their television preferences.
A computer generates 100 random numbers, and 100 people whose names correspond with the numbers on the list are chosen.
Use the following information to answer the next seven exercises. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new drug is currently under study to address a respiratory virus. It is given to patients once the patient exhibits symptoms of the virus. Of interest is the average (mean) length of time in days from the time the patient starts the treatment until the symptoms are alleviated. Two researchers each follow a different set of 40 patients with the respiratory virus from the start of treatment until the symptoms are alleviated. The following data (in days) are collected.
Researcher A: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34
Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29
Complete the tables using the data provided:
| Survival Length (in months) | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 0.5–6.5 | |||
| 6.5–12.5 | |||
| 12.5–18.5 | |||
| 18.5–24.5 | |||
| 24.5–30.5 | |||
| 30.5–36.5 | |||
| 36.5–42.5 | |||
| 42.5–48.5 |
| Survival Length (in months) | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 0.5–6.5 | |||
| 6.5–12.5 | |||
| 12.5–18.5 | |||
| 18.5–24.5 | |||
| 24.5–30.5 | |||
| 30.5–36.5 | |||
| 36.5-45.5 |
List two reasons why the data may differ.
Would you expect the data to be identical? Why or why not?
Suppose that the first researcher conducted his survey by randomly choosing one state in the nation and then randomly picking 40 patients from that state. What sampling method would that researcher have used?
Suppose that the second researcher conducted his survey by choosing 40 patients he knew. What sampling method would that researcher have used? What concerns would you have about this data set, based upon the data collection method?
Use the following data to answer the next five exercises: Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school. They collect the following data.
| Hours Played per Week | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 0–2 | 26 | 0.17 | 0.17 |
| 2–4 | 30 | 0.20 | 0.37 |
| 4–6 | 49 | 0.33 | 0.70 |
| 6–8 | 25 | 0.17 | 0.87 |
| 8–10 | 12 | 0.08 | 0.95 |
| 10–12 | 8 | 0.05 | 1 |
| Hours Played per Week | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 0–2 | 48 | 0.32 | 0.32 |
| 2–4 | 51 | 0.34 | 0.66 |
| 4–6 | 24 | 0.16 | 0.82 |
| 6–8 | 12 | 0.08 | 0.90 |
| 8–10 | 11 | 0.07 | 0.97 |
| 10–12 | 4 | 0.03 | 1 |
Give a reason why the data may differ.
Would the sample size be large enough if the population is school-aged children and young adults in the United States?
Researcher A concludes that most students play video games between four and six hours each week. Researcher B concludes that most students play video games between two and four hours each week. Who is correct?
As part of a way to reward students for participating in the survey, the researchers gave each student a gift card to a video game store. Would this affect the data if students knew about the award before the study?
Use the following data to answer the next five exercises: A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem-solving skills. Patients were asked to use the software program twice a day, once in the morning and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in Table 1.32. The second study collected the data in Table 1.33.
| Group | Showed improvement | No improvement | Deterioration |
|---|---|---|---|
| Used program | 142 | 43 | 15 |
| Did not use program | 72 | 110 | 18 |
| Group | Showed improvement | No improvement | Deterioration |
|---|---|---|---|
| Used program | 105 | 74 | 19 |
| Did not use program | 89 | 99 | 4 |
The first study was performed by the company that designed the software program. The second study was performed by the American Medical Association. Which study is more reliable?
The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement?
Patients who used the software were also a part of an exercise program whereas patients who did not use the software were not. Does this change the validity of the conclusions from Exercise 1.31?
Is a sample size of 1,000 a reliable measure for a population of 5,000?
A question on a survey reads: "Do you prefer the delicious taste of Brand X or the taste of Brand Y?" Is this a fair question?
Is it possible for two experiments to be well run with similar sample sizes to get different data?
1.3 Frequency, Frequency Tables, and Levels of Measurement
What type of measure scale is being used? Nominal, ordinal, interval or ratio.
- High school soccer players classified by their athletic ability: Superior, Average, Above average
- Baking temperatures for various main dishes: 350, 400, 325, 250, 300
- The colors of crayons in a 24-crayon box
- Social security numbers
- Incomes measured in dollars
- A satisfaction survey of a social website by number: 1 = very satisfied, 2 = somewhat satisfied, 3 = not satisfied
- Political outlook: extreme left, left-of-center, right-of-center, extreme right
- Time of day on an analog watch
- The distance in miles to the closest grocery store
- The dates 1066, 1492, 1644, 1947, and 1944
- The heights of 21–65 year-old women
- Common letter grades: A, B, C, D, and F
1.4 Experimental Design and Ethics
Design an experiment. Identify the explanatory and response variables. Describe the population being studied and the experimental units. Explain the treatments that will be used and how they will be assigned to the experimental units. Describe how blinding and placebos may be used to counter the power of suggestion.
Discuss potential violations of the rule requiring informed consent.
- People in a correctional facility are offered good behavior credit in return for participation in a study.
- A research study is designed to investigate a new children’s allergy medication.
- Participants in a study are told that the new medication being tested is highly promising, but they are not told that only a small portion of participants will receive the new medication. Others will receive placebo treatments and traditional treatments.