Pahntoms Redwood trees are the tallest plants on Earth. California is famous for its giant Redwood trees. But just how tall are they? A random sample of 47 California Redwood trees was taken and their heights measured. (This was not easy by the way.) The sample mean average height was 248 feet with a standard deviation of 26 feet. Does this data meet the assumptions necessary to perform a hypothesis test? If so, use a 5% significance level to test the claim that Redwood trees have an average height greater than 240 feet.

We conclude that the Redwood trees have an average height greater than 240 feet.

Step-by-step explanation:

We are given that a random sample of 47 California Redwood trees was taken and their heights measured. The sample mean average height was 248 feet with a standard deviation of 26 feet.

We have to test the claim that Redwood trees have an average height greater than 240 feet.

Let = mean weight bag filling capacity of machine.

SO, Null Hypothesis, : 240 feet   {means that the Redwood trees have an average height smaller than or equal to 240 feet}

Alternate Hypothesis, : > 240 feet   {means that the Redwood trees have an average height greater than 240 feet}

The test statistics that will be used here is One-sample t test statistics as we don't know about the population standard deviation;

                        T.S.  =  ~

where,  = sample mean average height = 248 feet

             s = sample standard deviation = 26 feet

             n = sample of trees = 47

So, test statistics  =    ~   

                               =  2.109

Now at 5% significance level, the t table gives critical value of 1.6792 at 46 degree of freedom for right-tailed test. Since our test statistics is higher than the critical value of t as 2.109 > 1.6792, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the Redwood trees have an average height greater than 240 feet.

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).