STAT 302 502 Week 11 11 6 Exam 2 Review Significance Tests 1 One population proportion p 2 One population mean with unknown Is the active ingredient concentration as stated on the label 325 mg tablet H 0 325 H a 325 Is nicotine content greater than the written 1 mg cigarette on average H 0 1 H a 1 Is the proportion of male births is less than 51 7 for this hospital H 0 0 517 p H a 0 517 p 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 76 sample SD s 5 SE 1 667 s n we would use the standard error for a sampling model of sample mean bc we don t know the population SD If is known then use Normal z scores If 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 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 is true p value is NOT the probability that H 0 P value if p value is LARGER than we fail to reject H if p value is SMALLER than we reject H 0 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 Since n is large the unknown can be replaced by the sample value s x z n x z s n 99 z 2 58 95 z 1 96 90 z 1 645 INTERPRETATION It is incorrect to say that there is a probability of 0 90 that is between 11 15 and 12 05 In fact this probability is either 1 or 0 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 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 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 we can either reject or fail to reject our hypotheses
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