Answer:
0.9826 = 98.26% probability that the sample proportion will differ from the population proportion by less than 0.03.
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.
For a proportion p in a sample of size n, we have that 
In this problem, we have that:

So


If 411 are sampled, what is the probability that the sample proportion will differ from the population proportion by less than 0.03?
This is the pvalue of Z when X = 0.07+0.03 = 0.1 subtracted by the pvalue of Z when X = 0.07 - 0.03 = 0.04. So
X = 0.1



has a pvalue of 0.9913
X = 0.04



has a pvalue of 0.0087
0.9913 - 0.0087 = 0.9826
0.9826 = 98.26% probability that the sample proportion will differ from the population proportion by less than 0.03.