Answer:
0.0264 = 2.64% probability that the proportion of wrong numbers in a sample of 448 phone calls would differ from the population proportion by more than 3%
Explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
and standard deviation
A telephone exchange operator assumes that 9% of the phone calls are wrong numbers.
This means that
Sample of 448
This means that
Mean and standard deviation:
What is the probability that the proportion of wrong numbers in a sample of 448 phone calls would differ from the population proportion by more than 3%?
More than 9% + 3% = 12 or less than 9% - 3% = 6%. Since the normal distribution is symmetric, these probabilities are the same, so we find one of them and multiply by 2.
Probability it is less than 6%
P-value of Z when X = 0.06. So
By the Central Limit Theorem
has a p-value of 0.0132
2*0.0132 = 0.0264
0.0264 = 2.64% probability that the proportion of wrong numbers in a sample of 448 phone calls would differ from the population proportion by more than 3%