4. Furnace repair bills are normally distributed, with a mean of 270 dollars and

Question

4. Furnace repair bills are normally distributed, with a mean of 270 dollars and a standard deviation of 25 dollars. We randomly select 100 of these repair bills.

(a) Describe the sampling distribution of the sample mean of furnace repair bills for a random sample of 100 bills. Explain why.

Conditions:

Shape:

Center:

Spread:

b) Find the probability that a randomly selected sample of 100 furnace repair bills has a mean cost of between 270 and 272 dollars.

Z-scores: ____________ Table Values: ____________ Final Answer: ____________ Concluding Sentence:

Explanation / Answer

(a) Describe the sampling distribution of the sample mean of furnace repair bills for a random sample of 100 bills. Explain why.

By central limit theorem,

Conditions: It is normaly distributed regardless of smaple size, because the original distirbution is already normal.

Shape: Bell shaped.

Center: $270 [same as population mean]
Spread: sigma(X) = sigma/sqrt(n) = 25/sqrt(100) = 2.5 [ANSWER]

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b) Find the probability that a randomly selected sample of 100 furnace repair bills has a mean cost of between 270 and 272 dollars.

We first get the z score for the two values. As z = (x - u) sqrt(n) / s, then as          
x1 = lower bound =    270      
x2 = upper bound =    272      
u = mean =    270      
n = sample size =    100      
s = standard deviation =    25      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u) * sqrt(n) / s =    0      
z2 = upper z score = (x2 - u) * sqrt(n) / s =    0.8      
[Z SCORES: 0 AND 0.8 [ANSWER]

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Using table/technology, the left tailed areas between these z scores is          
          
P(z < 0) =    0.5      
P(z < 0.8) =    0.7881 [ANSWER, TABLE VALUES]

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Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.2881 [FINAL ANSWER]

Hence, the probability that a randomly selected sample of 100 furnace repair bills has a mean cost of between 270 and 272 dollars is 0.2881. [CONCLUDING SENTENCE]

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