Download S - kellydoran

AP Statistics
I.
Inference Part 2
10-2: Inference for Two Means
Unit 9 Day 2
Confidence Intervals for the difference of two means (2-samp T-Int)
P – define parameter, population, and process
A – check Assumptions & Conditions
-Random – both samples must be random
-Normal – if the populations of each sample are Normal, then the sample distribution is Normal; If the
sample size is at least 30, then the sample distribution is Normal according to the Central Limit Theorem; if
the sample size is less than 30 and the population is not Normal, then you must draw TWO boxplots or graphs
to show there are no outliers or strong skew for either sample.
-Independent – for experiments, state independence if experiment was done properly; for samples,
check the 10% condition (add a 0 to each sample size and check population size).
S – Statistical Work (2-Samp T-Int)
-by hand: Formula: (𝑥̅1 − 𝑥̅2 ) ± 𝑡 ∗ √
𝑠1 2
𝑛1
+
𝑠2 2
Calculator:
𝑛2
𝑠1 2
*point estimate = (𝑥̅1 − 𝑥̅2 ); critical value = 𝑡 ∗ ; standard error = √
𝑛1
+
𝑠2 2
𝑛2
*margin of error = (critical value)(standard error)
*degrees of freedom – the calculator will find the exact degrees of freedom using a complicated formula
(found on page 637). When doing the problems by hand, find the critical value (𝑡 ∗ ) by taking the degrees
of freedom from the smallest sample size to be more conservative. However, always use the calculator and
report the degrees of freedom when constructing confidence intervals for two sample means.
S – Conclusion: “We are ____% confident that the interval between ____________ and ____________ (units) will
capture the true mean difference in parameter 1 and parameter 2 (in context).
Example 1) In a study of heart surgery, one issue was the effect of drugs called beta-blockers on the pulse
rate of patients during surgery. The available subjects were divided at random into two groups of 30
patients each. One group received a beta-blocker; the other group received a placebo. The pulse rate of
each patient at a critical point during the operation was recorded. The treatment group had a mean
pulse rate of 65.2 and standard deviation of 7.8. For the control group, the meal pulse rate was 70.3 and
the standard deviation was 8.3. Construct and interpret a 99% confidence interval for the mean difference
in pulse rates.
P – 2-sample T-interval
𝜇1 = true mean pulse rate of patients receiving beta-blockers; 𝜇2 = true mean pulse rate of
patients receiving the placebo
A -Random – subjects were assigned randomly to groups
-Normal – both sample sizes are 30, so the samples are approximately Normal according to the CLT.
N -Independent – random assignment into treatment groups ensures independence
S–
***Always choose “No” for Pooled***
df = 57.78
(-10.64, .439)
S – We are 99% confident the interval between -10.64 and .439 bpm will capture the true mean difference in
pulse rates between patients receiving beta-blockers during surgery and patients receiving placebos.
II.
Significance Tests for the difference of two means (2-sample T-Test)
P – state hypothesis, define parameters & populations, and identify procedure
Null: H0 : µ1 = µ2
*the null will ALWAYS be that there is no difference between treatments
Alternative: Ha : µ1 >/</≠ µ2 *choose the appropriate inequality symbol based on the problem
A – check Assumptions & Conditions & state alpha level
-the conditions are the same as for confidence intervals for 2 means
S – Statistical Work (2-Samp T-Test)
-by hand: Formula: 𝑡 =
̅1 −𝑥
̅2 )−0
(𝑥
2
2
𝑛1
𝑛2
𝑠
𝑠
√ 1 + 2
, then use tcdf to find p-value
-Calculator:
*degrees of freedom – the calculator will find the exact degrees of freedom using a complicated formula
(found on page 637). When doing the problems by hand, find the critical value (𝑡 ∗ ) by taking the degrees
of freedom from the smallest sample size to be more conservative. Always use the calculator and report
the degrees of freedom when running a significance test for two sample means. State t, p-value, & df
S – Conclusion:
Because our p-value of _______ is less than [greater than] α = 0.05, we [fail to] reject the null
hypothesis and [cannot] conclude ______________________________________________ (alternative in context).
**if the p-value is greater than alpha, then insert the words in brackets in the conclusion**
Example 2) Suppose we want to test the hypothesis that beta-blockers reduce mean pulse rate (scenario
from Example 1). Run an appropriate significance test at the α=0.05 level.
P
2-sample T-test
H0 : µ1 = µ2
Ha : µ1 < µ2
𝜇1 = true mean pulse rate of patients receiving beta-blockers
𝜇2 = true mean pulse rate of patients receiving the placebo
A – α=0.05
Random, Normal, & Independent were checked & verified in Example 1.
S–
***Always choose “No” for Pooled***
df = 57.78
t = -2.453
p-val = 0.0086
S – Because our p-value of 0.0086 is less than α = 0.05, we reject the null hypothesis and conclude that the
true mean pulse rate for patients who received beta blockers is lower than those who received a placebo.
Beta-blockers reduce mean pulse rate.
Notes for homework (pg. 652)


35 is a review from Chapter 6. Find the center, shape, and spread and then use that information to
find the probability.
The book uses the more conservative degrees of freedom when they do the calculations by hand,
but my answers online are by using the calculator. Make sure to state the degrees of freedom since
the answers vary depending on which one you use.