Question

Andy is a product manufacturer. He believes 71% of his customers are satisfied with the quality of his products. From a random sample of 100 customers, what are the following probabilities? (If necessary, use the sampling distribution of the sample proportion, NOT the normal approximation to binomial.)

(a) What is the probability that less than 72 customers are satisfied with the quality? (Round your answers to 4 decimal places, if needed.)
(b) What is the probability that at least 72 customers are satisfied with the quality? (Round your answers to 4 decimal places, if needed.)
(c) What is the probability that the sample proportion of customers who are satisfied with the quality is between 68% and 75%, inclusively?

Answer 1

To find probabilities related to customer satisfaction, we can use the sampling distribution of the sample proportion. By calculating cumulative probabilities, we can determine the likelihood of certain outcomes.

Step-by-step explanation:

The probabilities can be calculated using the sampling distribution of the sample proportion. Let's calculate these probabilities:

1. (a) To find the probability that less than 72 customers are satisfied with the quality, we need to calculate the cumulative probability from 0 to 71 customers. Using the sampling distribution of the sample proportion with p = 0.71 (probability of customer satisfaction), n = 100 (sample size), and x = 71 (number of customers satisfied), we can use a calculator or software to find this probability. It equals to 0.2611.
2. (b) To find the probability that at least 72 customers are satisfied with the quality, we need to calculate the cumulative probability from 72 to 100 customers (inclusive). Using the same sampling distribution, we can find this probability as 1 minus the cumulative probability from 0 to 71. It equals to 0.7389.
3. (c) To find the probability that the sample proportion of customers who are satisfied with the quality is between 68% and 75%, inclusively, we need to calculate the cumulative probability from 68 to 75. We can use the sampling distribution and calculate the individual probabilities for each value and then sum them up. The resulting probability equals to 0.6246.