Download Inferences

Statistics 3502/6304
Prof. Eric A. Suess
Chapter 5
Inferences about Population Central Values
• Estimation and Confidence Intervals
• Hypothesis Testing
Estimation of the Population Mean
• Point estimate of the population mean µ is the sample mean 𝑦.
• The sample mean is the “best guess” at the value of the unknown
population mean.
• How accurately does the sample mean estimate the population
mean?
• Anyone play darts?
Estimation of the Population Mean
• If we assume the population is Normally distributed with mean µ
standard deviation σ, then the Central Limit Theorem tells us that the
distribution of the sample mean 𝑦 is Normally distributed with mean
σ
µ standard deviation
(Called the standard error.)
𝑛
• In repeated sampling, about 95% of the sample means 𝑦 will be
within approximately 2 standard errors of the mean µ (More precisely
1.96 standard errors.)
• Draw a picture.
Estimation of the Population Mean
• The Central Limit Theorem gives.
𝑃(𝜇 − 1.96
𝜎
𝑛
≤ 𝑦 ≤ 𝜇 + 1.96
𝜎
)
𝑛
= .95
• Draw a picture.
• Or the 95% Confidence Interval for µ, when σ known.
𝑃(𝑦 − 1.96
𝜎
𝑛
≤ 𝜇 ≤ 𝑦 + 1.96
• Which can be written in general as
𝑦 ± 𝑧𝛼
2
𝜎
𝑛
𝜎
)
𝑛
= .95
Estimation of the Population Mean
• See Table 5.2 on page 229 for a z-values for different Confidence
Levels.
• When σ is unknown, it will be estimated using the sample standard
deviation s. See Page 250. Later.
Hypothesis Testing of the Population Mean
Hypothesis Testing
1. Research Hypothesis (also called the Alternative Hypothesis) 𝐻𝑎 :
2. Null Hypothesis, 𝐻0 :
3. Test Statistic
4. Rejection Region
5. Check assumptions and draw conclusions.
Hypothesis Testing of the Population Mean
• Examples:
• Blood Testing
• Jury trial
• Errors:
• Type I Error, False Positive, Convict an innocent prisoner
• Type II Error, False Negative, Release a guilty prisoner
• See Table 5.3 on page 234
Hypothesis Testing of the Population Mean
• One-tailed Test
• Example 5.5 page 235, 236
• Two-tailed Test
• Example 5.6 page 237
• Summary of Hypothesis Test for µ, when σ known. Page 238
Hypothesis Testing of the Population Mean
• Significance Level of the Test, α
• The significance level is usually set at 5%, limiting the Type I Error to
5%.
• The p-value is often computed using computer software and
compared to the significance level to determine if the Null Hypothesis
is Rejected
• Reject 𝐻0 : if the p-value ≤ 𝛼 = 0.05
Hypothesis Testing of the Population Mean
• The p-value is defined to be the probability of obtaining a value of the
test statistics that is as likely or more likely to Reject 𝐻0 : as the actual
observed value of the test statistic, assuming that the Null Hypothesis
is true.
• If the p-value is small then we will Reject 𝐻0 :
Hypothesis Testing of the Population Mean
• The Test Statistic when σ is know is:
𝑦−𝜇
𝑧=𝜎
𝑛
• The Test Statistic when σ is unknown is:
𝑦−𝜇
𝑡= 𝑠
𝑛
Hypothesis Testing of the Population Mean
• The z test statistic uses the z-table that we know. See Table 1 at the
start of the book, or Table 1 at the end of the book at the bottom.
• The t test statistic uses the t-table that we know. See Table 2 at the
back of the book.
Example
• Next time we will look at a number of examples of Confidence
Intervals and Hypothesis Testing.