lec19 7.3~7.4 -8.1

μ = .75, where μ is the true average inside diameter of a certain type of PVC pipe .
p < 0.10, where p is the proportion of defective circuit boards among all circuit boards produced by a certain manufacturer. μ1 – μ2 = 0, where μ1 and μ2 denote the true average breaking strengths of two different types of twine.
(n 1)S 2
2 /2,n1
2
(n 1)S 2
21 /2,n1
Proposition: A 100(1-α)% confidence interval for the variance σ2 of a normal population has lower limit
• The result of a hypothesis test is either: – The null hypothesis is rejected
• This is a strong result. It indicates that your alternative hypothesis has convincing data behind it.
2 2 n 1 s 2,n1
and upper limit
n 1 s
2
2 1 2, n1
A confidence interval for σ has lower and upper limits that are the square roots of the corresponding limits in the interval for σ2.
Tolerance Intervals（容许区间）
Let k be a number between 0 and 100. A tolerance interval for capturing at least k% of the values in a normal population distribution (which has unknown μ and σ) with a confidence level 95% has the form
16 137,324.3 16 137,324.3 , 76,172.3, 318, 064.4 28.845 6.908
Taking the square root of each endpoint yields (276.0,564.0) as the 95% CI for σ.
• Typical claims: – A new methodhe items meet (or do not meet) the specifications
– There is a difference in the average quality characteristic in the output of two processes
Example 7.15 The accompanying data on breakdown voltage of electrically stressed circuits was read from a normal probability plot. The straightness of the plot gave strong support to the assumption that breakdown voltage is approximately normally distributed . 1170 1510 1690 1740 1900 2000 2030 2100 2190 2200 2290 2380 2390 2480 2500 2580 2700 Let σ2 denote the variance of the breakdown voltage distribution and it is unknown. Determine the 95% confidence interval of σ2. Solution: The computed value of the sample variance is s2 = 137,324.3, the point estimate of σ2. With df = n-1 =16, a 95% CI require χ2.975,16 = 6.908 and χ2.025,16 = 28.845. The interval is
The CI for σ2 can be easily built from this statistic based on its distribution. The area under the χ2ν curve (v df) to the right of χ2α/2,ν is α/2, as is the area to the left of χ21-α/2,ν.Thus, the area captured between these two critical values is 1- α.
Summary of Chap. 7
• CI for μ of a normal distribution with known σ; general method for deriving CIs. • Large-sample CIs for μ and p of general distributions with known or unknown σ; one-sided CIs. • CIs for μ of a normal distribution with unknown σ (using t distribution); a PI for a single observation; • CIs for σ2 and σ of a normal distribution (using χ2 distribution).
x tolerance critical value s
Tolerance critical values for k = 90, 95, and 99 in combination with various sample sizes are given in Appendix Table A.6. This table also include critical values for a confidence level of 99%. Replacing by + gives an upper tolerance bound, and using – gives a lower tolerance bound. Critical values for obtaining these one-sided bounds also appear in Appendix Table A.6.
– The null hypothesis fails to be rejected
• This is a weak result.
• It DOES NOT imply that the null hypothesis is true. • Only that there is not a convincing amount of data to support the alternative • If you want to prove something, it should be your alternative hypothesis, not your null hypothesis.
χ2ν curve Each shaded area = α/2 1- α

2 1 /2,

2
/2,
Now ν = n -1,
2 (n 1) S 2 2 P 1 /2,n1 /2, n1 1 2
Rearrange the inequation, we obtain:
Definition
The null hypothesis （ 原 假 设 、 零 假 设 、 无 效 假 设 ） , denoted by H0, is the claim about one or more population characteristics that is initially assumed to be true (the “prior belief” claim). The alternative hypothesis（备择假设）, denoted by Ha, is the assertion that is contradictory to H0. The null hypothesis will be rejected in favor of the alternative hypothesis only if sample evidence suggests that H0 is false. If the sample does not strongly contradict H0, we will continue to believe in the truth of the null hypothesis.
Chap. 8 Tests of Hypotheses Based on a Single Sample
8.1 Hypotheses and Test Procedures
A statistical hypothesis（统计假设）, or hypothesis, is a claim either about the value of a single population characteristic or about the values of several population characteristics. Examples of hypotheses: