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Answer:
The probability that the mean battery life would be greater than 533.2 minutes (in a sample of 75 batteries) is
Step-by-step explanation:
The main thing we have to take into account in this question is that we are about to find the probability of a <em>sample mean</em>. The distribution for <em>sample means</em> follows a <em>normal distribution</em> with mean and standard deviation
. Mathematically
For values of the sample , no matter the distribution the data come from.
And the variable <em>z</em> follows a <em>standard normal distribution</em>, and, as we can remember, this distribution has a mean = 0 and a standard deviation = 1. Mathematically
[1]
That is
We have a variance of 3364. That is, a <em>standard deviation</em> of
The population mean is
The sample size is
The sample mean is
With all this information, we can solve the question
The probability that the mean battery life would be greater than 533.2 minutes
Using equation [1]
With this value of z we can consult a <em>cumulative standard normal table</em> (or use some statistic program) to find the cumulative probability for <em>z</em> (and remember that this variable follows a standard normal distribution).
Most standard normal tables have values for z for only two decimals, so we can round the previous value for z as z = 0.48.
Then
However, in the question we are asked for . As well as all normal distributions, the standard normal distribution is symmetrical around the mean, and we have
Thus
Rounding to four decimal places, we have
So, the probability that the mean battery life would be greater than 533.2 minutes is (in a sample of 75 batteries) .