STAT 302 - 502 - Week 11 - 11/6 Exam 2 Review Significance Tests: 1. One population proportion, p. 2. One population mean, µ, with unknown σ. ● Is the active ingredient concentration as stated on the label (325 mg/ tablet)? H0: µ = 325 Ha: µ ≠ 325 ● Is nicotine content greater than the written 1 mg/cigarette, on average? H0: µ = 1 Ha: µ > 1 ● Is the proportion of male births is less than 51.7% for this hospital? H0: p ≥ 0.517 Ha: p < 0.517 EX) For a random sample of 9 women, the average resting pulse rate is x = 76 beats per minute, and the sample standard deviation is s = 5. The standard error of the sample mean is: Answer: -sample mean = μ = 76 -sample SD = s = 5 SE =  = 1.667s√n -we would use the standard error for a sampling model of sample mean (bc we don’t know the population SD) ● If σ is known, then use Normal (z-scores). ● If σ is unknown: ○ If n is large, then use Normal (z-scores). ○ If n is small, then use t-distribution (t-values, df).SMALL SAMPLE T-TEST FOR A POPULATION MEAN (where σ is unknown): -if the sample is too small (n<30), we use the t-test (or distribution) -the LARGER the sample size, the LARGER the test statistic -the data is more likely to be statistically significant if it was based on a LARGE sample size than on a small one REMINDERS: -never ACCEPT the null hypothesis -set the significance level before performing the test (usually given) -base the hypotheses on the data given in the problem (not based on the sample data though) -p-value: is NOT the probability that H0 is true P-value: -if p-value is LARGER than α, we fail to reject H0 -if p-value is SMALLER than α, we reject H0 -don’t forget to double the p-value if you are performing a TWO-TAILED hypothesis test Confidence Intervals: A Confidence Interval is an interval of numbers containing the most plausible values for our Population Parameter. The probability that this procedure produces an interval that contains the actual true parameter value is known as the Confidence Level and is generally chosen to be 0.90, 0.95 or 0.99. LARGE SAMPLE CONFIDENCE INTERVAL FOR A POPULATION MEAN x ± z*√n Since n is large, the unknown σ can be replaced by the sample value s. x ± z*s√n 99%: z = 2.58 95%: z = 1.96 90%: z = 1.645INTERPRETATION: It is incorrect to say that there is a probability of 0.90 that μ is between 11.15 and 12.05. In fact this probability is either 1 or 0 (μ either is or is not in the interval). It is correct to say that the 90% refers to the percentage of all possible intervals that contain μ i.e. to the estimation process rather than a particular interval. It is also incorrect to say that 90% of all tutorials had between 11.15 and 12.05 missing students. -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