Statistical Inference: Confidence Intervals and Hypothesis Tests, Exams of Mathematical Statistics

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STAT 310 DISCUSSION 7

TA: Yi Chai Office: 1335N MSC Email: [email protected] Webpage: http://www.stat.wisc.edu/∼chaiyi Office Hours: 11:00-12:00pm T and 1:00-2:00pm Th or by appointment

1. Confidence Intervals for Proportions The conventional 95% confidence interval for p is:

ˆp ± 1. 96

√

pˆ(1 − pˆ) n

2. Hypothesis Tests
   • Testing
     • H 0 : μ = μ 0
     • Test statistic: z = x¯ − μ 0 σ/

√

n

• Alternative hypothesis and P -values Alternative Hypothesis P -values HA : μ 6 = μ 0 Pr{|Z| > |z|} HA : μ > μ 0 Pr{Z > z}
• Use standard normal when variance is known: Pr{|Z| > |z|} = 2Φ(−|z|)
• Use t distribution when variance is unknown: Pr{|Z| > |z|} = 2Ftn− 1 (−|z|)
• Interpreting a P-Value The smaller the p-value, the more inconsistent the data is with the null hypothesis, the stronger the evidence is against the null hypothesis in favor of the alternative.
• Comparing α and P-values
• The significance level α is a prespecified, arbitrary value, that does not depend on the data.
• The p-value depends on the data.
• Rejecting the null hypothesis when the p-value is less than the significance level α

3. Examples

• Example 1: (6.3.1 from the textbook.) Suppose measurements (in centimeters) are taken using an instrument. There is error in the measuring process and a measurement is assumed to be distributed N (μ, σ 02 ), where μ is the exact measurement and σ^20 = 0.5. If the following (n = 10) measurements 4.7, 5.5, 4.4, 3.3, 4.6, 5.3, 5.2, 4.8, 5.7, 5.3 were obtained, assess the hypothesis H 0 : μ = 5 by computing the relevant P -value. Also compute a 0.95-confidence interval for the unknown μ.
• Example 2: (6.3.2 from the textbook.) Suppose in Example 2 we drop the assumption that σ^20 = 0.5. Then assess the hypothesis H 0 : μ = 5 and compute a 0.95-confidence interval for the unknown μ.

• Example 3: (6.3.8 from the textbook.) A polling firm conducts a poll to determine what pro- portion θ of voters in a given population will vote in an upcoming election. A random sample of n = 250 was taken from the population, and the proportion answering yes was 0.62. Assess the hypothesis H 0 : θ = 0.65 and construct an approximate 0.90-confidence interval for θ.