Review for Test #3
ECON 205
CHAPTER 8 – CONFIDENCE INTERVALS
POINT ESTIMATE –
• estimate of a population parameter using a single number (a statistic from a
sample.
➢𝑥ҧ is the point estimate for µ (population mean)
➢𝑝Ƹ is the point estimate for 𝑝 (population proportion)
CONFIDENCE INTERVAL:
POINT ESTIMATE ± E
“E” is the maximal margin of error for a c% level of
confidence.
Formula:
E = (critical value)(Standard Error)
(the standard error is the standard deviation of a sampling distribution)
Two distributions for finding critical values:
1. Standard normal (z) distribution
2. Student’s 𝑡 distribution (more conservative)
as sample size increases the student’s 𝑡 distribution
approaches the standard normal (z) distribution.
CRITICAL VALUES for determining confidence intervals:
Find in distribution indicated:
❖For µ
➢ σ known – 𝑧𝑐 from standard normal distribution
➢ σ unknown – estimate with 𝑠 –𝑡𝑐 from student’s 𝑡 distribution
(with d.f. – n-1)
❖For 𝑝 𝑤ℎ𝑒𝑛 𝑛𝑝 > 5 𝑎𝑛𝑑 𝑛𝑞 > 5 - 𝑧𝑐 from standard normal
distribution
CALCULATING STANDARD ERROR
(standard deviation of sampling distribution)
Standard Error for µ
𝜎
• 𝑛
•
𝑠
𝑛
Standard Error for 𝑝
•
𝑝ො𝑞ො
𝑛
CONFIDENCE INTERVAL: Point Estimate ± E
𝜎
➢ For µ: 𝑥ҧ ± 𝑧𝑐
𝑛
or
𝑥ҧ ± 𝑡𝑐
➢ For 𝑝 ∶
𝑝Ƹ ± 𝑧𝑐
𝑠
𝑛
𝑝ො 𝑞ො
𝑛
In the design stages of statistical research projects, it is
a good idea to decide in advance on the confidence
level you wish to use and to select the maximal margin
of error 𝐸 you want for your project. If determine that 𝐸
needs to be smaller than preliminary study:
solution: increase 𝑛
𝑛 is in the denominator for all standard errors; if 𝑛
increases the standard error decreases – interval will be
narrower or shorter
CHAPTER 9
Hypothesis testing
Null Hypothesis: makes a claim about a population
parameter, in this chapter 𝜇 or 𝑝
In the null hypothesis statement always state the parameter
about which claim is made and use the = sign.
➢𝐻𝑜 : 𝜇 = 𝑘
➢𝐻𝑜 : 𝑝 = 𝑘
ALTERNATE HYPOTHESIS STATEMENT (NEVER use = sign)
➢𝐻1 : 𝜇 > 𝑘
𝐻1 : 𝑝 > 𝑘
➢𝐻1 : 𝜇 < 𝑘
𝐻1 : 𝑝 < 𝑘
➢𝐻1 : 𝜇 ≠ 𝑘
𝐻1 : 𝑝 ≠ 𝑘
SET UP TEST:
: (the level of significance of the test)
1.Null Hypothesis: state population parameter making claim
about
2.Alternate Hypothesis: state alternate using <, >, or ≠
3.State level of significance of test , α (alpha) – probability of
committing a Type I error
4.Use corresponding sample statistic to test the claim; calculate
sample test statistic
5.Conclude test with decision using critical region method or
p-value method
p-value method
• If p-value ≤ α
REJECT NULL HYPOTHESIS
• If p-value > α
FAIL TO REJECT NULL HYPOTHESIS
Critical Region Method
LEFT TAIL TEST:
Reject region
Fail to reject region
RIGHT TAIL TEST:
Fail to reject region
Reject region
TWO TAIL TEST:
Reject region
Reject region
Fail to reject region
ALWAYS:
SAME CONCLUSION CRITICAL REGION METHOD
OR P-VALUE METHOD
CHAPTER 4 – CORRELATION AND SIMPLE LINEAR REGRESSION
study of relationship between variables
Sample correlation coefficient 𝑟
-1.0 ≤ 𝑟 ≤ 1.0
The closer 𝑟 is to 1 (positive or negative) the strong the
relationship
The equation of the least squares line calculated from
sample data is:
If calculated positive 𝑟 then 𝑏 will be positive
If calculated negative 𝑟 then 𝑏 will be negative
Correlation and Regression
Example
Expected positive slope - correlation coefficient corroborates positive. Moderate correlation.