The basal diameter of a sea anemone is an indicator of its a

The basal diameter of a sea anemone is an indicator of its age. The density curve shown here represents the distribution of diameters in a certain large population of anemones; the population mean diameter is 4.2 cm, and the standard deviation is 1.4 cm.4 Le Lets (Y DASH ) represent the mean diameter of 25 anemones randomly chosen from the population.

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a) Find the approximate value of pr{ 4<=Y DASH <=5}

mean diameter of 25 anemones randomly chosen from the population. 4 10 Diameter (cm) (a) Find the approximate value of Pr(4s Ys5) (b) Why is your answer to part (a) approximately cor- rect even though the population distribution of diameters is clearly not normal? Would the same approach be equally valid for a sample of size 2 rather than 25? Why or why not?

Solution

a)

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    4      
x2 = upper bound =    5      
u = mean =    4.2      
n = sample size =    25      
s = standard deviation =    1.4      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    -0.714285714      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    2.857142857      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.237525262      
P(z < z2) =    0.997862633      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.760337371   [ANSWER]

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b)

This is so because of the central limit theorem, where it states that the distribution of the sample means becomes closer to the normal distribution as the sample size gets larger (here, n = 25).

If the sample size is just 2, then we cannot impose the central limit theorem, as the sample size is not big enough to do that approximation.