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Statistics 400 - Lecture 9 Today: Sections 8.3 Read 8.3 and 8.4 for next day VERY IMPORTANT SECTIONS!!! Confidence Intervals for the Mean Last day, introduced a point estimator…a statistic that estimates a population parameter Often more desirable to present a plausible range for the parameter, based on the data We will call this a confidence interval Ideally, the interval contains the true parameter value In practice, not possible to guarantee because of sample to sample variation Instead, we compute the interval so that before sampling, the interval will contain the true value with high probability This high probability is called the confidence level of the interval Confidence Interval for for a Normal Population Situation: Have a random sample of size n from N ( , ) Suppose value of the standard deviation is known Value of population mean is unknown Last day we saw that 100(1 )% of sample means will fall in the interval: z , z /2 /2 n n Therefore, before sampling the probability of getting a sample mean in this interval is (1 ) Equivalently, P z / 2 X z / 2 (1 ) n n Equivalently, P X z / 2 X z / 2 (1 ) n n The interval below is called a 100(1 )% confidence interval for X z , X z /2 /2 n n Example To assess the accuracy of a laboratory scale, a standard weight known to be 10 grams is weighed 5 times The reading are normally distributed with unknown mean and a standard deviation of 0.0002 grams Mean result is 10.0023 grams Find a 90% confidence interval for the mean Interpretation What exactly is the confidence interval telling us? Consider the interval in the previous example. What is the probability that the population mean is in that particular interval? Consider the interval in the previous example. What is the probability that the sample mean is in that particular interval? Large Sample Confidence Interval for Situation: Have a random sample of size n (large) Suppose value of the standard deviation is known Value of population mean is unknown If n is large, distribution of sample mean is Can use this result to get an approximate confidence interval for the population mean When n is large, an approximate for the mean is: 100(1 )% confidence interval Large Sample Confidence Interval for (unknown standard deviation) Situation: Have a random sample of size n (large) Suppose value of the standard deviation is unknown Value of population mean is unknown When n is large, replacing the population standard deviation with the sample estimate gives a good approximation When n is large and the population standard deviation is unknown, an approximate 100(1 )% confidence interval for the mean is: Example (8.19) Amount of fat was measured for a random sample of 35 hamburgers of a particular restaurant chain Sample mean and sample standard deviation were found to be 30.2 and 3.8 grams Find a 95% confidence interval for the mean fat content of hamburgers for this chain Changing the Length of a Confidence Interval Can shorten the length of a confidence interval by: Using a difference confidence level Increasing the sample size Reducing population standard deviation Hypothesis Testing Hypothesis testing is a statistical technique to test if a conjecture about a population parameter is true Has 4 Main Steps: Null and Alternate Hypotheses Test Statistic P-Value Decision based on pre-specified error rate Example Heights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches Company claims taking 500 mg of Vitamin C makes the girls taller 1. Hypotheses Begin by making an assumption of no change This statement is called the null hypothesis (H0) Test will be designed to assess evidence against H0 Hypothesis we suspect is true is called alternate hypothesis (H1) Assume H0 is true, collect data and see if there is evidence against H0 and in favor of H1 Example Heights of one-year-old girls normally distributed with mean 30 inches and standard deviation of 1.2 inches Company claims taking 500 mg of Vitamin C makes the girls taller H0: H1: