The amounts of nicotine in a certain brand of cigarette are normally distributed

Question

The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.896 g and a standard deviation of 0.318 g. The company that produces these cigarettes claims that it has now reduced the amount of nicotine. In what range would you expect to find the middle 95% of amounts of nicotine in these cigarettes (assuming the mean has not changed)? Between ___________ and ______________ .

If you were to draw samples of size 56 from this population, in what range would you expect to find the middle 95% of most average amounts of nicotine in the cigarettes in the sample? Between__________ and ____________ .

Enter your answers as numbers. Your answers should be accurate to 4 decimal places.

Is there a way to do this in excel???

Explanation / Answer

Mean ( u ) =0.896
Standard Deviation ( sd )=0.318
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a.
P ( Z < x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is -1.96
P( x-u/s.d < x - 0.896/0.318 ) = 0.025
That is, ( x - 0.896/0.318 ) = -1.96
--> x = -1.96 * 0.318 + 0.896 = 0.2727                  
P ( Z > x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is 1.96
P( x-u/ (s.d) > x - 0.896/0.318) = 0.025
That is, ( x - 0.896/0.318) = 1.96
--> x = 1.96 * 0.318+0.896 = 1.5193                  
95% of amounts of nicotine in these cigarettes b/w 0.2727, 1.5193
b.
Mean ( u ) =0.896
Standard Deviation ( sd )= 0.318/ Sqrt(n) = 0.0425
Number ( n ) = 56
Normal Distribution = Z= X- u / (sd/Sqrt(n) ~ N(0,1)                  
P ( Z < x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is -1.96
P( x-u/s.d < x - 0.896/0.0425 ) = 0.025
That is, ( x - 0.896/0.0425 ) = -1.96
--> x = -1.96 * 0.0425 + 0.896 = 0.8127                  
P ( Z > x ) = 0.025
Value of z to the cumulative probability of 0.025 from normal table is 1.96
P( x-u/ (s.d) > x - 0.896/0.0425) = 0.025
That is, ( x - 0.896/0.0425) = 1.96
--> x = 1.96 * 0.0425+0.896 = 0.9793                  
95% of amounts of nicotine in these cigarettes b/w 0.8127, 0.9793

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