As part of a promotion for a new type of cracker, free samples are offered to shoppers in a local supermarket. The probability that a shopper will buy a packet of crackers after tasting the free sample is 0.200. Different shoppers can be regarded as independent trials. Let be the proportion of the next n shoppers that buy a packet of the crackers after tasting a free sample. How large should n be so that the standard deviation of is no more than 0.01?

We need n being at least 1600.

Step-by-step explanation:

The Central Limit Theorem estabilishes 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, the mean is and the standard deviation is

The probability that a shopper will buy a packet of crackers after tasting the free sample is 0.200

This means that

How large should n be so that the standard deviation of is no more than 0.01?

s at most 0.01, so a sample size of at least n when s = 0.01

We need n being at least 1600.

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