1. A random sample of 15 values of PAR (photosynthetically active radiation) wer

Question

1.              A random sample of 15 values of PAR (photosynthetically active radiation) were taken at noon in a certain forest (in moles per meter2 per second) :

                                    492      621      521      561      518

                                    571      594      629      603      608

                                    538      562      546      532      576

Assume that PAR is approximately normally distributed with a known standard deviation of 40 moles/(m2s).

a.What is the mean value observed for PAR?

b.What is the standard error of the mean value of PAR, given this sampling scenario?

c.Find the 95% confidence intervals for the unknown population mean of the PAR values and interpret its meaning.

d.Find the 90% confidence intervals for the unknown population mean of the PAR values and interpret its meaning.

e.Compare the two intervals.

f.Test the hypothesis that average PAR value is 550 moles/(m2s), at the =0.05 level. (Be sure to right down all steps – including hypotheses – as in the lecture notes, and interpret the meaning of the test!)

Please show me step by step the formulas used to get these anawers.

Explanation / Answer

a)

Using technology,

X = sample mean =    564.8 [ANSWER]

*****************

b)

Using technology,

s = sample standard deviation =    40

As

n = sample size =    15

Then

Standard error = s/sqrt(n) = 40/sqrt(15) = 10.32795559 [ANSWER]

*******************

c)

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    564.8          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    40          
n = sample size =    15          
              
Thus,              
              
Lower bound =    544.557579          
Upper bound =    585.042421          
              
Thus, the confidence interval is              
              
(   544.557579   ,   585.042421   ) [ANSWER]

We are 95% confident that the true mean of PAR is between 544.56 and 585.04.

***********************

d)

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.05          
X = sample mean =    564.8          
z(alpha/2) = critical z for the confidence interval =    1.644853627          
s = sample standard deviation =    40          
n = sample size =    15          
              
Thus,              
              
Lower bound =    547.8120248          
Upper bound =    581.7879752          
              
Thus, the confidence interval is              
              
(   547.8120248   ,   581.7879752   ) [ANSWER]

We are 90% confident that the true mean of PAR is between 547.81 and 581.79. [ANSWER]

************************

e)

As we can see, the two intervals has the same center, but the 95% confidence interval is wider due to the larger critical z.

*******************************************

Hi! Please submit the next part as a separate question. That way we can continue helping you! Please indicate which parts are not yet solved when you submit. Thanks!

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.