Discussion Exercise 8 Spring 2009 Name Matthew Marck Section 0101 Confidence Intervals and Hypothesis Testing Begin by reviewing what we know about the Sampling Distribution of Sample Means Answer the following questions Imagine a distribution of all textbook prices at our school This distribution has a mean of 40 and a standard deviation of 10 Now imagine that all possible samples of 25 textbooks are taken from this population with replacement and for each sample the sample mean x is calculated These sample means are plotted to display their distribution Show what you think this distribution of sample means would look like by 1 a In the space below sketch what you think the shape pf the sampling distribution for the means of all possible samples of 25 textbook prices would look like u 40 b Write the value for the mean of this distribution and show where it would be located in c What would the value of 1 standard deviation of the sampling distribution of sample the sampling distribution means be equal to x sqrt n 10 sqrt 25 2 d Calculate the actual values of the mean 1 2 and 3 standard deviations Mark and label points on the x axis arrow to show where these values occur Now add the values for the mean 1 2 and 3 standard deviations X 1 x 42 2 x 44 3 x 46 1 x 38 2 x 36 3 x 34 e What assumptions did you make when you drew this picture I assumed that the sample mean was the same as the population mean and that it is normally distributed If you draw one sample at random from the population of textbook prices what will f be the probability of obtaining a sample mean more extreme than the mean or 1 2 standard deviations Z 42 40 2 z 1 0 3413 0 5 0 3413 0 1587 Z 44 40 2 z 2 0 4772 0 5 0 4772 0 0228 Confidence Intervals Simulations Go to http onlinestatbook com stat sim conf interval index html Click Begin and read CAREFULLY what the simulation is showing when you click Sample Change the parameters used when sampling and try different values to help you answer the questions below 2 How well did you simulation capture the true population mean a It seems to look as if the mean holds true for as a whole individuals varied a What percent of the 99 confidence intervals were successful b What percent of the 95 confidence intervals were successful a 97 a 94 3 Which is wider a 95 confidence interval or a 99 confidence interval 99 confidence is wider 4 Assume you create a 95 confidence interval based on n 25 and separately create a 95 confidence interval based on n 5 Does a larger sample size increase the probability that your interval includes the true population mean value YES or NO 5 You take a sample of size 30 and calculate both the 95 confidence interval and also the 99 confidence interval Is it possible to capture the true population mean in the 99 confidence interval but miss it in the 95 confidence interval Yes in certain cases it is I can see this on the simulation the true mean lies in the blue on a few 6 When you construct a 95 confidence interval what are you 95 confident about a that the interval includes the sample mean b that the interval includes the population mean c that the interval includes 95 of the data Questions about Hypothesis Testing 7 You are examining tulip bulbs and are interested in if the mean weight of tulip bulbs is more than 10 grams Circle the correct alternative hypothesis a Ha 10 b Ha 10 c Ha d Ha 10 10 Ho would be 10grams 8 Water used for cooling is discharged from a power plant into a river You take 50 water samples of this discharge water It has been determined that as long as the mean temperature of the discharged water is at most 150 F there will be no negative effects on the river s ecosystem You will use your water samples to investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above 150 F The resulting data will be used to test the hypotheses H0 150 F versus Ha 150 F In the context of this example a True or F A Type A Correct Decision would conclude that the mean temperature of water being discharged is 150o F or less when this is in fact true b True or F A Type I Error would mean concluding that the water being discharged from the power plant has a mean temperature in excess of 150o F when in fact the mean water temperature is not greater than 150o F