Hypothesis Testing & Confidence Interval: yield of corn

Bushels of Corn
Crop researchers plant 15 plots with a new variety of corn. The yields in bushels per acre are:

138.0 139.1 113.0 132.5 140.7 109.7 118.9 134.8
109.6 127.3 115.6 130.4 130.2 111.7 105.5
Assume that the standard deviation of the population is known to be σ = 10 bushels per acre

QUESTIONS:

1. What is xthe standard deviation of xˉˉ ?

2. Find the 90%, 95%, and 99% confidence interval for the mean yield for this variety of corn.

3. How do the margin of error in #2 change as the confidence level increases?

Now suppose that the crop researchers obtained the same value of xˉˉ from a sample of 60 plots rather than 15.

1. What is xCompute the 95% confidence interval for the mean yield 

2. Is the margin of error larger or smaller than the margin of error found for the sample of 15 plots? Explain in ONE SENTENCE why the change occurs.

3. Will the 90% and 99% intervals for a sample of size 60 be wider or narrower than those for
n = 15?

4. How large a sample is required to estimate the mean yield within ±4 bushels per acre with
90% confidence?

Suppose the mean yield of corn in the United States is about 120 bushels per acre. A survey of 40 farmers this year gives a sample mean yield of xˉˉ = 123.8 bushels per acre. We want to know whether this is good evidence that the national mean this year is not 120 bushels per acre. Assume that the farmers surveyed are an SRS from the population of all commercial corn growers and that the standard deviation of the yield in this population is 10 bushels per acre. Are you convinced that the population mean is not 120 bushels per acre? Use α = 1%.

1. State Ho and Ha.

2. Calculate the test statistic and give the P-value.

3. Give a 99% confidence interval for the mean bushels per acre and state whether the
confidence interval includes .

4. State your conclusion based on α = 1%. Would your answer change at 5% significance level?
Why or why not?