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fomenos
3 years ago
9

The lifetime of a certain type of battery is normally distributed with mean value 10 hours and standard deviation 1 hour. There

are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages?
Mathematics
1 answer:
Oksi-84 [34.3K]3 years ago
4 0

Answer:

z=1.64

And if we solve for a we got;

10 +1.64*1=11.64

So the value of height that separates the bottom 95% of data from the top 5% is 11.64  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent certain lifetime of a population, and for this case we know the distribution for X is given by:

X \sim N(10,1)  

Where \mu=10 and \sigma=1

For this part we want to find a value a, such that we satisfy this condition:

P(X>a)=0.05   (a)

P(X   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.95 of the area on the left and 0.05 of the area on the right it's z=1.64. On this case P(Z<1.64)=0.95 and P(z>1.64)=0.05

If we use condition (b) from previous we have this:

P(X  

P(z

But we know which value of z satisfy the previous equation so then we can do this:

z=1.64

And if we solve for a we got;

10 +1.64*1=11.64

So the value of height that separates the bottom 95% of data from the top 5% is 11.64  

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