Statistics Lecture 9

zToday: Sections 8.3 zRead 8.3 and 8.4 for next day zVERY IMPORTANT SECTIONS!!!

Confidence Intervals for the Mean zLast day, introduced a point estimator…a statistic that estimates a population parameter zOften more desirable to present a plausible range for the parameter, based on the data zWe will call this a confidence interval

zIdeally, the interval contains the true parameter value zIn practice, not possible to guarantee because of sample to sample variation zInstead, we compute the interval so that before sampling, the interval will contain the true value with high probability zThis high probability is called the confidence level of the interval

Confidence Interval for for a Normal Population zSituation: yHave a random sample of size n from ySuppose value of the standard deviation is known yValue of population mean is unknown

zLast day we saw that of sample means will fall in the interval: zTherefore, before sampling the probability of getting a sample mean in this interval is zEquivalently,

z The interval below is called a confidence interval for

Example zTo assess the accuracy of a laboratory scale, a standard weight known to be 10 grams is weighed 5 times zThe reading are normally distributed with unknown mean and a standard deviation of grams zMean result is grams zFind a 90% confidence interval for the mean

Interpretation zWhat exactly is the confidence interval telling us? zConsider the interval in the previous example. What is the probability that the population mean is in that particular interval? zConsider the interval in the previous example. What is the probability that the sample mean is in that particular interval?

Large Sample Confidence Interval for zSituation: yHave a random sample of size n (large) ySuppose value of the standard deviation is known yValue of population mean is unknown

zIf n is large, distribution of sample mean is zCan use this result to get an approximate confidence interval for the population mean zWhen n is large, an approximate confidence interval for the mean is:

Large Sample Confidence Interval for (unknown standard deviation) zSituation: yHave a random sample of size n (large) ySuppose value of the standard deviation is unknown yValue of population mean is unknown

zWhen n is large, replacing the population standard deviation with the sample estimate gives a good approximation zWhen n is large and the population standard deviation is unknown, an approximate confidence interval for the mean is:

Example (8.19) zAmount of fat was measured for a random sample of 35 hamburgers of a particular restaurant chain zSample mean and sample standard deviation were found to be 30.2 and 3.8 grams zFind a 95% confidence interval for the mean fat content of hamburgers for this chain

Changing the Length of a Confidence Interval zCan shorten the length of a confidence interval by: yUsing a difference confidence level yIncreasing the sample size yReducing population standard deviation

Hypothesis Testing zHypothesis testing is a statistical technique to test if a conjecture about a population parameter is true zHas 4 Main Steps: yNull and Alternate Hypotheses yTest Statistic yP-Value yDecision based on pre-specified error rate

Example zHeights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches zCompany claims taking 500 mg of Vitamin C makes the girls taller

1. Hypotheses zBegin by making an assumption of no change zThis statement is called the null hypothesis (H 0 ) zTest will be designed to assess evidence against H 0 zHypothesis we suspect is true is called alternate hypothesis (H 1 ) zAssume H 0 is true, collect data and see if there is evidence against H 0 and in favor of H 1

Example zHeights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches zCompany claims taking 500 mg of Vitamin C makes the girls taller zH 0 : zH 1 :