A bottle maker calculates that 13% of his bottles are defective. If the bottle maker is accurate, what is the probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%? Round your answer to four decimal places

0.2420 = 24.20% probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%.

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

The standard deviation of the binomial distribution is:

Normal probability distribution

Problems of normally distributed samples can be 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.

When we are approximating a binomial distribution to a normal one, we have that , .

In this problem, we have that:

So

What is the probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%?

12% of 454 is 0.12*454 = 54.48.

The number of defective bottles is a discrete value, so you can only have 54 or 55 defective bottles. 54 is less than 12%, so this probability is the pvalue of Z when X = 54.

has a pvalue of 0.7580

1 - 0.7580 = 0.2420

0.2420 = 24.20% probability that the proportion of defective bottles in a sample of 454 bottles would be less than 12%.