Answer:
Using the normal probability distribution, with a capacity of 350, it is enough for all abused on 90.82% of nights.
274 shelters will be needed.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
If the city’s shelters have a capacity of 350, will that be enough places for abused women on 95% of all nights?
What is the percentile of 350?
This is the pvalue of Z when X = 350.
has a pvalue of 0.9082.
Using the normal probability distribution, with a capacity of 350, it is enough for all abused on 90.82% of nights.
If not, what number of shelter openings will be needed?
The 95th percentile, which is X when Z = 1.645. So
274 shelters will be needed.