1 Hypothesis Tests on the Mean H 0 :  =  0 H 1 :    0.

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Presentation transcript:

1 Hypothesis Tests on the Mean H 0 :  =  0 H 1 :    0

2 Hypothesis Tests (one side) H 0 :  =  0 H 1 :  >  0 H 0 :  =  0 H 1 :  <  0

3 Example: two-sided test Suppose Guido takes a random sample of n=25 and obtains an average burn rate of 51.3 cm/s. Specs require that burn rate must be 50 cm/s, and the standard deviation is known to be 2 cm/s. He decides to specify a type I error probability (significance level) of What conclusions can be drawn?

4 P-value The P-value is the smallest level of significance that would lead to rejection of the null hypothesis H0 with the given data.

5 Probability of Type II Error The probability of the type II error is the probability that Z 0 falls between -z  /2 and z  /2 given that H 1 is true.

6  Requirement Sample Size

7 Example (cont.) Suppose Guido wants to design the burn rate test so that if the true mean burn rate differs from 50 cm/s by as much as 1 cm/s, the test will detect this (i.e. reject the null hypothesis) w.p Determine the sample size required to detect this departure.

8 Confidence Intervals

9 Example Construct a 95% confidence interval for the burn rate. 25 samples were taken. The sample mean is 51.3 and the standard deviation is 2. What is the relationship between hypothesis testing and confidence intervals?

10 Error Bound Sample Size

11 Example Suppose that we wanted the error in estimating the mean burn rate of the rocket propellant to less than 1.5 cm/s, with 95% confidence. What is the required sample size?

12 One-Sided CIs

13 General CIs