Statistical Methods
Statistical Question
A question that can be answered by collecting data where there is variability in the data.
Example.
How old are the members of the senior class?
Confidence Interval
statistical range that estimates the true value of a population parameter (such as a mean or proportion) with a specified level of confidence, indicating the precision and reliability of the sample estimate
Observational Study
researchers observe and record data for the subjects in the sample (they do NOT assign treatments). Can only show correlation.
Example.
What percent of students at this university are involved in at least one extracurricular activity?
Alternative Hypothesis
The claim to be tested; the opposite of the null hypothesis.
Example.
a hypothesis that there is a significant difference between populations influenced by a non-random cause
Population (in stats)
all members of a defined group
Example.
Every doctor in America
Statistical Experiment
Trials of an event that has more than one probability-based outcome.
Example.
A coin toss is a statistical experiment because it has predictable outcomes that are determined by chance.
Control Group
a group of specimens in an experiment to which no change is made; does not receive the experimental treatment
Example.
In an experiment investigating the effect of talk therapy on alleviating depression, the group receiving only the medicine would be the control group.
Survey
A list of questions used to gather information from a sample in order to make a statement or conclusion about the population.
Two-Sided Alternative Hypothesis
When the alternative hypothesis states that the parameter is different from the null value (if it could be either greater than or less than).
Margin of Error (in stats)
a number that represents how far above or below the actual mean may be from the experimental mean in a study
Sample
a subset of the population
Example.
The 250 doctors polled in a survey
Simple Random Sample / Random Sample
a sample in which each member of the subset has an equal probability of being chosen
Example.
Every member is assigned a number and a random number generator is used to select the sample.
Experimental Investigation
Researchers assign subjects in the sample to certain treatments, then observe the effects of the treatment. Can show causation (cause and effect).
Example.
Does using algebra tiles during instruction help freshman students learn how to solve equations?
Null Hypothesis
A statement about the value of a population parameter, such as the population mean (µ) or the population proportion (p).
Example.
a hypothesis that there is no significant difference between populations
One-Sided Alternative Hypothesis
When the alternative hypothesis states that a parameter is greater than the null value or if it states that the parameter is less than the null value.
Jessica wants to survey parents about their thoughts on the new cafeteria menu. What is the best way to obtain a valid sample?
A. Email the students and ask them to ask their parents to fill out a survey.
B. Ask people at the grocery store if they have children in school. If they do, ask them to fill out a survey.
C. Get a list of parents from the school and randomly send 10 percent of them the survey.
D. Get a contact list of from the PTA of parents involved in the PTA and send them survey questions.
C. Get a list of parents from the school and randomly send 10 percent of them the survey.
Explanation:
This is the best choice because it should include the entire population of parents.
The counselor of a school would like to survey the students about the lunch program the school offers. Which of the following methods would be the best way to collect the data?
A. Survey the first 100 students that arrive at school.
B. Survey the last 100 students that go through the lunch line.
C. Survey 100 students who bring their lunch to school.
D. Survey 100 students, chosen by random student ID numbers.
D. Survey 100 students, chosen by random student ID numbers.
Explanation:
A random sample best avoids bias in a survey. Since each student has a unique student ID number, 100 students can be chosen at random.
A middle school cafeteria director wants to survey students about new menu options. Which of the following survey options will effectively yield the most valid information about all students' opinions?
A. Use a random name selector that sources names from a list of all enrolled students.
B. Ask all students to fill out a form during lunch with their friends.
C. Survey every 5th student in the lunch line for all three grade levels.
D. Survey 5th period math classes.
A middle school cafeteria director wants to survey students about new menu options. Which of the following survey options will effectively yield the most valid information about all students' opinions?
A. Use a random name selector that sources names from a list of all enrolled students.
Explanation:
This method ensures the most random sample that will likely represent student opinion.
Sasha is analyzing statistical data and computes a confidence interval with a sample size of 200. What will happen to the confidence interval if Sasha changes the sample size to 100?
A.The confidence interval will become half as big if the sample size decreases.
B. The confidence interval will widen if the sample size decreases.
C. The confidence interval will become narrower if the sample size decreases.
D. The confidence interval will double if the sample size decreases.
B. The confidence interval will widen if the sample size decreases.
Explanation:
A larger sample size will more accurately represent the population and therefore have a smaller margin of error and a smaller confidence interval.
A study conducted on 100 mice showed that 54%\pm 10%54 of mice enjoy eating watermelon. The researchers report the results with 95% confidence. What is the margin of error in this study?
10%
Explanation:
The margin of error shows the degree of uncertainty in the study results and is represented by \pm x\%±x%.
A biologist took a random sample of 200200 American black bears to study the weight of the adult bear. The sample mean was 149.7149.7 and the sample standard deviation was 23.823.8. When calculated, the 90\%90% confidence interval for the mean weight was (147, 152)(147,152) lbs.
Which of the following statements is the correct interpretation of the 90\%90% confidence interval?
A. If the biologist kept taking samples, 90\%90% of the sample intervals would overlap the true mean of the data.
B. There is a 90 percent probability that the true mean weight is between 147 and 151 pounds.
C. With repeated sampling, a sample mean between 147147 and 151151 pounds is produced in about 90\%90% of samples.
D. 90\%90% of the bears weighed between 147147 and 151151 pounds.
A. If the biologist kept taking samples, 90\%90% of the sample intervals would overlap the true mean of the data.
Explanation:
We are 90\%90% confident that the stated interval captured the true mean, therefore with repeated sampling, the intervals produced will encompass the population mean in about 90\%90% of samples.
Which of the following will reduce the width of a confidence interval?
I. Increasing the confidence level
II. Increasing the sample size
III. Decreasing the standard deviation
II and III only
Explanation:
Increasing the sample size would decrease the standard error, which would decrease the confidence interval. Moreover, as sample size increases, the standard deviation decreases, so II and III are essentially the same thing.