Answer:
and if we round up to the nearest integer we got n =174
Step-by-step explanation:
Previous concepts
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
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".
Let X the random variable who represents the test score of a student taking his final examination. We know from the problem that the distribution for the random variable X is given by:
From the central limit theorem we know that the distribution for the sample mean is given by:
Solution to the problem
We want to find the value of n that satisfy this condition:
And we can use the z score formula given by:
And we have this:
And we can express this like this:
And by properties of the normal distribution we can express this like this:
If we solve for we got:
Now we can find a quantile on the normal standard distribution that accumulates 0.03 of the area on the left tail and this value is:
And using this we have this equality:
If we solve for we got:
And then and if we round up to the nearest integer we got n =174