Vocabulary/
Concepts
Concepts
This distribution is used instead of the normal distribution when the population standard deviation is unknown and the sample size is small.
T-Distribution
Name 3 inference conditions
Random
10%
Normal/Large Sample
A random sample of 25 students has a mean SAT Math score of 540 with a sample standard deviation of 40. Construct a 95% confidence interval for the true mean SAT Math score.
bar x ± t^∗(s/sqrt n)=540±t^∗(8),
with df=24df = 24df=24, giving an interval of approximately (523.5, 556.5)?
Parameter for two-sample t
μ1 − μ2
Increasing the confidence level will make a confidence interval do this.
What is become wider?
This value, calculated as n−1, determines the exact shape of a t-distribution.
Degrees of Freedom (df)
What ensures independence when sampling without replacement?
10% Condition
A 90% confidence interval for the mean time it takes students to complete a test is (42.1, 47.9) minutes. Interpret this interval in context.
“We are 90% confident that the true mean time it takes students to complete the test is between 42.1 and 47.9 minutes”
μ1 = average hours studied by AP Stats weekly by Mr. Denny's 4th period class
μ2 = average hours studied by AP Stats weekly by Mr. Denny's 1st period class
Interpret the Confidence Interval (3,10) for μ1-μ2.
μ1 larger since positive interval, therefore Mr. Denny's 4th period must study more on average.
This happens to the confidence interval width when the sample size increases (holding confidence level constant).
What is it becomes narrower?
As the sample size (n) increases, the t-distribution begins to look more and more like this distribution.
What is the standard normal (z) distribution?
Why does the Random condition matter?
It allows us to create generalizations to the population of interest from which we have drawn the sample.
A teacher compares test scores from two classes.
Class A:
bar x_1 = 78, s_1 = 6, n_1 = 20
Class B:
barx_2 = 74, s_2 = 5, n_2 = 22
Construct a 95% confidence interval for the difference in population means
mu_1 - mu_2
.
(bar x_1 - bar x _2) +- t^* sqrt( (s_1^2)/n_1 + (s_2^2)/n_2 ) = (78-74) +- t^* sqrt(36/20 + 25/22)
,
giving an interval approximately (1.0, 7.0)
Define paired data
Two values per individual/pair
This quantity measures how far the endpoints of a confidence interval are from the sample mean.
What is the margin of error?
Compared to the normal distribution, the t-distribution has these, reflecting more variability in small samples.
What are thicker tails or (more area in the tails)
Name the 3 ways, in order, that you can satisfy your Normal/Large Sample condition
1. Samples were randomly selected from an approximately normal distributed population
2. Sample Size (n) is greater than 30
3. The data must be charted using a chart that shows shape (dot plot, histogram, boxplot, stem & leaf plot, etc.) and must show and include the phrase "No signs of strong skewness and no outliers"
A 95% confidence interval for
mu_(("online) - mu_(("person")
homework completion time is (–4.2, –0.6). Interpret this result.
We are 95% confident that students in the online class complete homework between 0.6 and 4.2 minutes faster on average than students in the in‑person class
________ is the parameter for Two Sample T-Interval and _______ is the parameter for Paired Data
μ1-μ2
μdiff
Effect of quadrupling sample size (n)?
Margin of Error cut in half
This calculator function will help you find the t-statistic. Give the appropriate inputs as well
InvT(one-side tail area, df)
Under what circumstances will the 10% condition not apply. Give and example.
When random sampling is not conducted like in the case of an medical experiment that involves volunteers.
A 95% confidence interval for the difference in mean quiz scores between two teaching methods is (–1.5, 3.8). What conclusion can be drawn about a difference in means
Because 0 is contained in the interval, there is no convincing evidence of a difference in the mean quiz scores between the two teaching methods”
In a two sample T -Interval, what does an interval containing "0" represent?
There is a plausible chance that there is no difference between the sample means..
Margin of Error does account for ______________ but does not account for ______________
Sampling Variability , Bias