Confidence Intervals of the Mean 1. You collected some ecological samples in a field experiment in west Texas. Collected were soil samples to determine the amount of nutrients that would need to be applied to sustain a desired level of crop growth. The observation was the amount of nutrient the soil contained. Samples were taken from 100 different locations. It was found that the nutrient data are normally distributed. The mean value of the samples was 16 units (the exact name for the units do not really matter in this problem, so just call them units). The sample standard deviation of the samples is 3.5 units.

a. estimate the probability range for the Confidence Interval of the Mean (Eq. 3.15) (95% confidence) for the parameters given above.

b. What is the interpretation of the Confidence Interval of the Mean?

c. Consider the experiment was performed again in a second year, however, this time only 15 sites were sampled. The mean value in year two was found to be 14.5 units and the sample standard deviation was found to be 2.2 units. Using equation 3.16, estimate the probability ranges for the 90%, 95%, and 99% confidence intervals of the mean. Hint: you will need to use the tinv function in MATLAB. You will need to estimate THREE probability ranges to answer this question. One range for 90%; one range for 95%, and one range for 99%.

d. Describe how the probability range changes as the confidence interval increases.

e. What is the probability range for the 50% confidence?

f. If you were to repeat the experiment described initially (100 samples per year) for twenty years, what percentage of the years would you expect the true mean value to be bracketed by the 50% confidence interval? Additionally, what percentage of the years would you expect the true mean value to be bracketed by the 95% confidence interval