From the data we collected the population mean height all my

From the data we collected, the population mean height (all my classes) was 68.3 inches with a standard deviation of 3.7 inches. Consider this to be the population. Describe the sampling distribution of the sample mean if we take a sample size of 31 students. Show all three. Find the probability that a randomly selected student will more than 69 inches tall Find the probability that a random sample of 31 students have a mean height of more than 69 inches. Interpret, in a sentence, number 3 above, using the context of the problem.

Solution

1.

By central limit theorem:

a) It is approximately normally distributed.

b) It has the same mean, u(X) = 68.3 in.

c) It has a standard deviation of sigma/sqrt(n) = 3.7/sqrt(31) = 0.664539617 in.

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2.

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    69      
u = mean =    68.3      
          
s = standard deviation =    3.7      
          
Thus,          
          
z = (x - u) / s =    0.189189189      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.189189189   ) =    0.42497227 [ANSWER]

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3.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    69      
u = mean =    68.3      
n = sample size =    31      
s = standard deviation =    3.7      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    1.053360825      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   1.053360825   ) =    0.146087826 [ANSWER]

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4.

The probability that a sample of 31 students will have a mean of 69 in or more is 0.146088.