Statistics Lecture 22

Last Day…completed 5.1 Today Parts of Section 5.3 and 5.4

Example Government regulations indicate that the total weight of cargo in a certain kind of airplane cannot exceed 330 kg. On a particular day a plane is loaded with 81 boxes of a particular item only. Historically, the weight distribution for the individual boxes of this variety has a mean 3.2 kg and standard deviation 1.0 kg. What is the distribution of the sample mean weight for the boxes? What is the probability that the observed sample mean is larger than 3.33 kg?

Statistical Inference deals with drawing conclusions about population parameters from sample data Estimation of parameters: Estimate a single value for the parameter (point estimate) Estimate a plausible range of values for the parameter (confidence intervals) Testing hypothesis: Procedure for testing whether or not the data support a theory or hypothesis

Point Estimation Objective: to estimate a population parameter based on the sample data Point estimator is a statistic which estimates the population parameter

Suppose have a random sample of size n from a normal population What is the distribution of the sample mean? If the sampling procedure is repeated many times, what proportion of sample means lie in the interval:

In general, 100(1-  )% of sample means fall in the interval Therefore, before sampling the probability of getting a sample mean in this interval is

Could write this as: Or, re-writing…we get:

The interval below is called a confidence interval for Key features: Population distribution is assumed to be normal Population standard deviation, , is known

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 grams Mean result is 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 confidence interval for the mean is:

Example Amount of fat was measured for a random sample of 35 hamburgers of a particular restaurant chain It is known from previous studies that the standard deviation of the fat content is 3.8 grams Sample mean was found to be 30.2 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

Sample Size for a Desired Width Frequent question is “how large a sample should I take?” Well, it depends One to answer this is to construct a confidence interval for a desired width

Sample Size for a Desired Width Width (need to specify confidence level) Sample size for the desired width

Example Limnologists wishes to estimate the mean phosphate content per unit volume of a lake water It is known from previous studies that the standard deviation is fairly stable at around 4 ppm and that the observations are normally distributed How many samples must be sampled to be 95% confidence of being within.8 ppm of the true value?

Example A plant scientist wishes to know the average nitrogen uptake of a vegetable crop A pilot study showed that the standard deviation of the update is about 120 ppm She wishes to be 90% confident of knowing the true mean within 20 ppm What is the required sample size?