Developmental specialists have observed three-year-old children playing with eac

Question

Developmental specialists have observed three-year-old children playing with each other in a standardized laboratory situation for one hour. During observation, a specialist will count the number of times each child speaks to another children. It is known that for all children the number of times one child will speak to another child during a 1 hour interval is normally distributed with a population mean of 12 with a population standard deviation of 4. During an observational study, a developmental specialist observed 40 three-year-old children playing in a standardized laboratory situation. During the observation, the specialist counted the number of times each child spoke to another child. The result, over several observations, is that the children spoke to one another an average of 10 times per hour of play, with a standard deviation of 3. (a) For the given scenario, what is the probability (as a decimal) that one randomly selected child will talk to other children less than 23 times in a given hour? Do not round on any in-between step. Only round your final answer. Round to four decimal places. Formula: Z:=:rac{X:-:mu}{sigma} (b)For the given scenario, what is the probability (as a decimal) that the average number of times 40 randomly selected children will talk to other children less than 11 times in a given hour? Round your final answer to four decimal places. Formula: Z:=:rac{overline{X}-mu}{rac{sigma}{sqrt{n}}}

Explanation / Answer

a)

Note that we us ethe population mean and standard deviation instead of the sample. u = 12, sigma = 4.

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    23      
u = mean =    12      
          
s = standard deviation =    4      
          
Thus,          
          
z = (x - u) / s =    2.75      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   2.75   ) =    0.997020237 [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    11      
u = mean =    12      
n = sample size =    40      
s = standard deviation =    4      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -1.58113883      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.58113883   ) =    0.056923149 [ANSWER]

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