Probability: What Affects Estimates

Spring 2010 Math 263 Deb Hughes Hallett

Math 263: Excel Assignment 8 on Confidence Intervals

You must do the computer work for this assignment yourself, although you may certainly talk to other people. Answers which appear to be copied will be treated as an integrity violation.

At the Department of Forestry and Natural Resources, Purdue Universiy, Maher Tadros is studying plants that may grow well in the desert of his native country, Jordan, where the climate is similar to that of southern Arizona. To be suitable for commercial development as food, plants should have high nitrogen content. The file Plants-Nitrates.xlsx contains the percentage of nitrate in 252 observations of Tadros’s plants.1 We consider these 252 observations the population and we assume that the data is normally distributed.

1. First we will study a single sample drawn from this population. Look at the OneSample page. (a) What is the mean of the population? (b) What is the standard deviation of the population? (c) What is the size of the sample that was selected? (d) In which cell is the mean of the sample? (e) In which cell is the standard deviation of the sample? (f) What math formula was used to calculate the left end of the confidence interval? (Click on the cell to see what Excel formula was input. What math formula does it represent?) (g) What math formula was used to calculate the right end of the confidence interval? (Click on the cell to see what Excel formula was input. What math formula does it represent?) (h) What is the confidence level of the confidence interval? How do you know? (i) If the population mean is outside the confidence interval, what must be true of the endpoints of the confidence interval? (j) How does what is in Cells P4 and P5 relate to your answer to part (i) ? (Press F9 to see the results from other samples.) (k) How does what is in Cell Q6 relate to your answer to part (j) ?

1 Reported in Introduction to the Practice of Statistics, Moore, MacCabe, Craig, 6th edn, Freeman, 2009 1 Spring 2010 Math 263 Deb Hughes Hallett

2. Now we look at the ManySamples page. One sample of size 10 has been selected. (a) In which cell is the mean of this sample? (b) In which cells are the endpoints of the 95% confidence interval of this sample? (c) What does cell S10 tell you? When does it show a 0 and when does it show a 1?

Create 100 samples of size 10 and their corresponding 95% confidence intervals by dragging down cells D10 to S10 until you have 100 samples. Press F9 to see that it is live.

(d) How many of your confidence intervals do not contain the population mean? How many do contain the population mean, ? (e) From theory, how many of your set of 100 confidence intervals do you expect not to contain the population mean ? (f) By pressing F9 twelve times, create twelve sets of 100 samples. For each set of 100, record how many do not contain the population mean, in a table like this one:

Set of 100 1 2 3 4 5 6 7 8 9 10 11 12 Number not containing

3. Changing the confidence level to 99% in the ManySamples page (a) How would you alter the formulas in cells Q10 and R10 to create a 99% confidence interval? (You don’t need to actually do it.) (b) From theory, in a set of one hundred 99% confidence intervals, how many do you expect not to contain the population mean, ? Changing the confidence level to 90% in the ManySamples page

(c) How would you alter the formulas in cells Q10 and R10 to create a 90% confidence interval? (You don’t need to actually do it.) (d) From theory, in a set of one hundred 90% confidence intervals, how many do you expect not to contain the population mean, ?