Probability

[Bigman00]

Hi there, this is my first time posting here. I'm having some trouble understanding what this question is asking me to start off with.

The nicotine content in a single cigarette of brand X has a distribution with a mean of 0.8mg and a standard deviation of 0.1mg. If 100 of these cigarettes are analyzed, what is the probability that the resulting sample mean nicotine content is more than 0.78mg?

I've gone over all my notes and I can't seem to find the formula our professor laid out for us, I know I should know this formula, if anybody can help me get started I'm sure the rest will come back to me.

 

[galactus]

This is just a basic normal distribution problem.

Use \(\displaystyle z=\frac{(x-\mu)\sqrt{n}}{\sigma}\)

Use your given info to find z. Look up the corresponding probability in a z table.

Note, the problem asks for "more than .78"

Thus, you will want the region to the right of the z value you find above.

Actually, with n=100, you could use the Empirical Rule to get very close. Notice that .78 is 2 standard deviations below the mean.

Just count up the area from 2 S.D left of the mean going right.

 

[ionisk]

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This is just a basic normal distribution problem.Use Use your given info to find z. Look up the corresponding probability in a z table.Note, the problem asks for "more than .78"Thus, you will want the region to the right of the z value you find above.Actually, with n=100, you could use the Empirical Rule to get very close. Notice that .78 is 2 standard deviations below the mean.Just count up the area from 2 S.D left of the mean going right.

 

[galactus]

\(\displaystyle z=\frac{(.78-.8)\sqrt{100}}{.1}=-2\)

Now, see the -2 at the bottom of the normal curve I posted?.

Add up the percentages to the left of the -2 and subtract from 1. 1-(.0015+.0235)=.975

Or, look up -2 in the z-table, see what the corresponding value is in the body of the table. Subtract that value from 1.

This is because the z table extends from the left and goes right.

The value in the table corresponding to z=-2 is .0228

Subtract from 1 and get 1-.0228=.9772

See?. They are not that far apart.

This means that the probability the sample mean nicotine content is higher than .78 mg is 97.72%.

 

[Bigman00]

Ahhh I see. Thank you very much for clearing this up for me, I had two different formulas written in my notebook, it was driving me mad trying to understand them. The way you laid it out though makes much more sense especially with the attachment. Just to verify, either way 1-(.0015+.0235) or looking -2 in the z-table, finding it's value and subtracting from 1 will work? I want to familiarize myself with all possible ways to look at this problem and problems like it