Suppose we have a population whose proportion of items with the desired attribute is p = 0: 5. (a) if a sample of size 200 is taken, find the probability that the proportion of successes in the sample will be between 0.47 and 0.51. draw a sketch to you solve this problem and use proper probability notation. (b) if a sample of size 100 is taken, find the probability that the proportion of successes in the sample will be between 0.47 and 0.51. draw a sketch to you solve this problem and use proper probability notation.

a. If a sample of size 200 is taken, the probability that the proportion of successes in the sample will be between 0.47 and 0.51 is 41.26%.

b. If a sample of size 100 is taken, the probability that the proportion of successes in the sample will be between 0.47 and 0.51 is 30.5%.

Step-by-step explanation:

This problem should be solved with a binomial distribution sample, but as the size of the sample is large, it can be approximated to a normal distribution.

The parameters for the normal distribution will be

We can calculate the z values for x1=0.47 and x2=0.51:

We can now calculate the probabilities:

If a sample of size 200 is taken, the probability that the proportion of successes in the sample will be between 0.47 and 0.51 is 41.26%.

b) If the sample size change, the standard deviation of the normal distribution changes:

We can calculate the z values for x1=0.47 and x2=0.51:

We can now calculate the probabilities:

If a sample of size 100 is taken, the probability that the proportion of successes in the sample will be between 0.47 and 0.51 is 30.5%.

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