1. : An alternative representation of level of significance. - normal distribution applies. - α level of significance (e.g. 5% in two tails) determines the rejection region, which means - in the rejection region sample means are far enough away from the assumed population mean. - that only 5% of sample means would fall in the rejection region by chance. - Then we can have 95% confidence that a random sample mean will fall in that interval called. - The confidence level is 1- α. (e.g.1-5%=95%) 1.96-1.960 f(z) Statistical Inference for the Mean Confidence Interval

2. Confidence interval of sample means: z 1 =1.96z 1 =-1.960 f(z) e.g. 5% level of significance in two tails, i.e. 95% confidence level. 2.5% Φ(<z1)=0.025 z1=-1.96 Φ(>z1)=0.025 z1=+1.96 Then the 95% confidence interval is Statistical Inference for the Mean Confidence Interval

3. Confidence interval of sample means: z 1 =1.96z 1 =-1.960 f(z) 2.5% Then the confidence interval is where z is determined by the confidence level. Statistical Inference for the Mean Confidence Interval

4. Confidence interval of population mean when it’s unknown: z 1 =1.96z 1 =-1.960 f(z) e.g. 5% level of significance in two tails,i.e. 95% confidence level. 2.5% Φ(<z1)=0.025 z1=-1.96 Φ(>z1)=0.025 z1=+1.96 Then the interval estimate for the population mean with 95% confidence is Statistical Inference for the Mean Confidence Interval

5. Confidence interval of population mean when it’s unknown: z 1 =1.96z 1 =-1.960 f(z) 2.5% Then the confidence interval is where z is determined by the confidence level. Statistical Inference for the Mean Confidence Interval

6. Calculations for both test of hypothesis and confidence limits by the methods have three requirements: - The sample must be random and representative. - The distribution of the variable must be a normal distribution, at least to a good approximation. If the sample contains enough observations, the sample mean will be normally distributed even though the original observations were not. ( ) -The standard deviation of the observation must be known reliably, probably from previous information. Statistical Inference for the Mean Confidence Interval