STAT 302 - 502 - Week 10 - 10/30 Type I Error: -we reject H0 but H0 is actually true Type II Error: -we do not reject H0 but H0 is actually false -we want to make sure that we are not "accepting" H0 when H0 is false -we also want to make sure that we are rejecting H0 when H0 is false t-distributions: -when we are using t-distributions, σ (population standard deviation) is unknown and we are also assuming that the population is ~Normal -for t-distributions we use: -we use the standard error because we do not know the population SD -after calculating the t-score, we use df = n-1 and the confidence level (usually 95%) and refer to the t-distribution table to find the p-value -when we are using t-distributions, we want to make sure that we are using df = n-1 instead of just using the sample size n -sometimes, the p-value will be between a range of numbers (ie: 0.02<p-value<0.01) -based on whether the p-value is greater than or less than α, we can either reject or fail to reject our hypotheses REMEMBER: -the p-value is not the probability that H0 is true -the p-value refers to the sample statistic, not the population parameter -p-value is the probability of observing a sample statistic that is extreme given that H0 is true Making Conclusions: -if p-value is LARGER than α, we fail to reject H0 -if p-value is SMALLER than α, we reject H0Exercise The p-value for a two-sided test of the null hypothesis, H0: μ = 10 was found to be 0.06. Therefore, using the same data, -Confidence Level of 95%: α = 0.05 p = 0.06 > 0.05 Therefore, the data is not statistically significant and we fail to reject our null hypothesis. This is because the confidence interval would contain the value 10 (if our null hypothesis was true) and getting a value of 10 is not that extreme. -Confidence Level of 90%: α = 0.10 p = 0.06 < 0.10 Therefore, the data is statistically significant and we reject our null hypothesis. This is because the confidence interval would not contain the value 10 and it was really rare for us to get a sample value of 10.Exercise Suppose that a manufacturer is testing one of its machines to make sure that the machine is producing more than 97% good parts (H0: p = 0.97 and HA: p > 0.97). The test results in a P-value of 0.12. In reality, the machine is producing 99% good parts. What probably happens as a result of our testing? Answer: -We failed to reject the null hypothesis because the data was not statistically significant. -But in actuality, the null hypothesis is false and the alternative hypothesis is true and by failing to reject a FALSE H0, we made a Type II error.Exercise Suppose that a device advertised to increase a car’s gas mileage actually does not work. We test it on a fleet of cars (with H0: not effective), and our data results in a P-value of 0.004. With 5% level of significance, what probably happens as a result of our experiment? Answer: -The p-value (0.004) was smaller than the significance level (0.05) -We rejected the null hypothesis because the data was statistically significant. -However, the device doesn’t actually work and by rejecting a TRUE H0, we made a Type I