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Suppose a botanist grows many individually potted eggplants, all treated identically and arranged in groups of four pots on the greenhouse bench. After 30 days of growth, she measures the total leaf area Y of each plant. Assume that the population distribution of Y is approximately normal with mean = 800cm^2 and SD = 90cm^2.

a) What percentage of the plants in the population will have leaf area between 750cm^2 and 850cm^2?

b) Suppose each group of four plants can be regarded as a random sample from the population. What percentage of the groups will have a group mean leaf area between 750cm^2 and 850cm^2

I believe that the final answer for a) should be 0.4215, and for b) should be 0.7335 but I have no idea how to calculate it.

Help please!

TooTone
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juknee
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2 Answers2

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Answer:

Part I

$\mu = 800$ and $\sigma = 90$

$$P(750<=X<=850) = P(\frac{(750-800)}{90} <= z <= \frac{(850-800)}{90})$$

$$P(750<=X<=850) = P(-.555<=z<=.555) = 0.710743 - 0.289257 = 0.421485$$

Part II

$\sigma = 90/\sqrt(4)$

$$P(750<=X<=850) = P(\frac{(750-800)}{(90/\sqrt(4)} <= z <= \frac{(850-800)}{(90/\sqrt(4)})$$

$$P(750<=X<=850) = P(-1.111<=z<=1.111) = 0.86674 - 0.13326 = 0.733479$$

Satish Ramanathan
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So by "believe" you mean "the back of the book says"? Anyway, you can find probabilities for normally distributed things using your normal-distribution table, which takes a $z$ score $z_0$ and gives you $P(z < z_0)$ in return. Turn each of your measurements into $z$-scores, and look them up.

(Note that to find the area between, you will need to find the difference between two areas, since all the areas in the table are to the left.)

tabstop
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