Suppose that the amount of chlorophyll near Stearns Wharf in Santa Barbara, Cali

Question

Suppose that the amount of chlorophyll near Stearns Wharf in Santa Barbara, California varies due to the seasons, the entry of fresh water into the arca, and the salt water from the ocean. A random sample of 16 days-was selected and the chlorophyll near Stearns Wharf was measured lin g/L) tor each day resulting in a sample mean of 2.93 g/L and afsample standard deviation or 2,D ug/L 6. (a) Calculate a 95% confidence interval to estimate the true mean amount of chlorophvll near Stearns Wharf 15 pointsl Answer (Please give each endpoint.) (b) What must be true in order for the conlidence interval in Part (a) to give a valid estimate? Be sponii3 points (c) Refer back to the story and to the confidence interval found in Part (a). Answer each of the following by circling your answer. 13 points cach A 95% confidence interval based on a smaller sample of 10 days would be A 99% confidence interval based on the original sample of 16 days would be . L onger Shorter Longe Shorter

Explanation / Answer

given that
mean = 2.93
n =16
and sd = 2.1

we know that the confidence interval is given as

mean +- z*sd/sqrt(n)

also fromn the z table the value of 95% CI is 1.96

putting the values

2.93+- 1.96*2.1/sqrt(16)

1.901 and 3.595

b)
for this to be true , the data must come from the normal distribution

c)
we know that CI is
mean +- z*sd/sqrt(n)
where margin of error is

z*sd/sqrt(n) , which is a function of Z and n
as z increases - MOE increases , as n increases moe decreases

for 95% the MOE would have a factor of 1.96/sqrt(10) = 0.6198

for 99% and n = 16
the moe is 0.645

so when MOE is more the width increases

so in both the cases it would be longer than what we observe is part A

1.96/sqrt(16) = 0.45

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