At the end of the school event the organizers estimated that

At the end of the school event the organizers estimated that a family of participants spent in average $100.00 with a standard deviation of $20.00.

What’s the probability that the mean amount spent will be between $90 and $115?      

Participants whose spending was in the 95% will receive a 20% off coupon on the tickets of the following event. What is the minimum spent amount that you should have in order to receive such a discount?

What is the maximum amount that you can spend and still be in the lowest 5%?              

How likely (what is the probability) is it to have spent an amount below $110?

How likely (what is the probability) is it that a family spent more than $80?         

How likely (what is the probability) is it to have spent between $90 and $105?  

If 36 participants (36 = size of the sample) are selected randomly, what’s the likelihood that their mean spent amount will be within $7 of the population mean? (mean +/- 7)       

Solution

At the end of the school event the organizers estimated that a family of participants spent in average $100.00 with a standard deviation of $20.00.
What’s the probability that the mean amount spent will be between $90 and $115?   

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    90      
x2 = upper bound =    115      
u = mean =    100      
          
s = standard deviation =    20      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -0.5      
z2 = upper z score = (x2 - u) / s =    0.75      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.308537539      
P(z < z2) =    0.773372648      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.464835109   [ANSWER]

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Participants whose spending was in the 95% will receive a 20% off coupon on the tickets of the following event. What is the minimum spent amount that you should have in order to receive such a discount?

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.95      
          
Then, using table or technology,          
          
z =    1.644853627      
          
As x = u + z * s,          
          
where          
          
u = mean =    100      
z = the critical z score =    1.644853627      
s = standard deviation =    20      
          
Then          
          
x = critical value =    132.8970725   [ANSWER]

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What is the maximum amount that you can spend and still be in the lowest 5%?

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.05      
          
Then, using table or technology,          
          
z =    -1.644853627      
          
As x = u + z * s,          
          
where          
          
u = mean =    100      
z = the critical z score =    -1.644853627      
s = standard deviation =    20      
          
Then          
          
x = critical value =    67.10292746   [ANSWER]

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How likely (what is the probability) is it to have spent an amount below $110?

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    110      
u = mean =    100      
          
s = standard deviation =    20      
          
Thus,          
          
z = (x - u) / s =    0.5      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   0.5   ) =    0.691462461 [ANSWER]

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