Study Design
This type of study involves minimal interaction between a researcher and those being observed.
Observational Study
Which type of central tendency measures the highest frequency in appearances of a certain point in a data set?
Mode (Most Common)
A teacher gives a math test to her class. The test scores are normally distributed with a mean of 70 and a standard deviation of 10. A student scores 85 on the test.
What is the z-score for the student's test score?
(85−70)/10 = 15/10 = 1.5
The heights of adult men in a certain country are normally distributed with a mean of 70 inches and a standard deviation of 3 inches.
What percentage of men are shorter than 67 inches?
15.87%.
Would a confidence interval of 80%, 90%, or 95% have a wider range?
95%
This type of data collection method is typically used when an experiment may be too expensive, time consuming, or risky.
Simulation
Find the mean:
[12, 15, 22, 29, 35].
12+15+22+29+35=113
Mean = 113/5 = 22.6
In a large university, the heights of male students are normally distributed with a mean of 68 inches and a standard deviation of 3 inches.
What is the z-score of a male student who is 72 inches tall?
[72 − 68]/3 = 4/3 = 1.333
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
What percentage of the population has an IQ score between 85 and 115?
68.26%
A 88% confidence level means that if you were to repeat the sampling process 1000 times, approximately how many of those intervals would contain the true parameter?
1000*0.88 = 880
Convenience Sampling
Find the median:
[8, 3, 7, 5, 10, 4, 6, 9, 12, 15].
Median = (7+8)/2 = 15/2 = 7.5
Two students, Alice and Bob, take different standardized tests. Alice scores 600 on a test with a mean of 500 and a standard deviation of 100. Bob scores 75 on a test with a mean of 70 and a standard deviation of 5.
Who performed better relative to their test group?
Neither/Both.
Both Alice and Bob have the same z-score of 1.0, so they performed equally well relative to their respective test groups.
The weights of newborn babies in a hospital are normally distributed with a mean of 7 pounds and a standard deviation of 1.2 pounds.
What percentage of newborns weigh more than 8.5 pounds?
10.56%.
A sample of 25 students is taken from a large university, and the mean GPA of the sample is 3.2 with a standard deviation of 0.4.
Construct a 95% confidence interval for the mean GPA of the entire student population.
3.2 +/- 0.1568 =
[3.0432, 3.3568]
Why is randomness so important in sampling?
Randomness ensures that each member of the population has an equally likely chance of being selected to participate in the study.
A school is analyzing the test scores of 25 students in a final exam. The scores are as follows:
43, 46, 48, 49, 49, 50, 50, 50, 50, 52, 55, 58, 60, 60, 60, 62, 62, 62, 64, 67, 70, 70, 71, 100, 100.
The school wants to report a summary of the performance to the district office.
Should the school use the mean, median, or mode to best represent the central tendency of the students' test scores? Justify your choice.
The median because the presence of the two 100s as outliers affects the mean score.
A student’s z-score on a standardized test is 2.5. The test has a mean score of 100 and a standard deviation of 15.
What was the student’s original test score?
The student's original test score is 137.5.
The lifespan of a particular brand of light bulb is normally distributed with a mean of 1000 hours and a standard deviation of 100 hours.
If a sample contains 75 lightbulbs, at least how many should last between 850 and 1150 hours?
65
DAILY DOUBLE!
A factory produces light bulbs with a known standard deviation of 100 hours. A random sample of 50 bulbs has a mean lifespan of 1200 hours.
Construct a 99% confidence interval for the mean lifespan of all light bulbs produced by the factory.
1200 +/- 36.43 =
[1163.57, 1236.43]
Which type of bias occurs when a researcher asks "leading questions" or "loaded questions" during a survey?
Response Bias
A teacher records the following test scores for 10 students in her class:
[65, 72, 78, 80, 83, 85, 90, 92, 95, 98.]
Calculate the Mean Absolute Deviation (MAD) of these test scores, as well as the Std Dev and the Variance.
MAD = [18.8+11.8+5.8+3.8+0.8+1.2+6.2+8.2+11.2+14.2]/10 = 81.8/10 = 8.18
Std Dev = 9.87
Var = 97.56
A math test was administered to a group of students, and the scores are normally distributed. The mean score on the test was 75. One student, Ashley, received a score of 90, and her z-score was 1.5.
What is the standard deviation of the test scores?
1.5 = [90−75]/σ
σ = 15/1.5
σ = 10
A standardized test has scores that are normally distributed with a mean of 500 and a standard deviation of 120.
If 2,000 students take this test, how many should score between 480 and 680?
998
FINAL JEOPARDY:
Three friends—Anna, Ben, and Chloe—are discussing their ages. Each friend knows the following about the others:
Based on these clues, what are the ages of Anna, Ben, and Chloe?
Anna: 15 years old
Ben: 10 years old
Chloe: 5 years old