bookstore claims that 50% of its customers are satisfied with
the service and prices.
If this claim is true, what is the probability that in a random sample of 600 customers less
than 45% are satisfied with services and price?
To find the probability that in a random sample of 600 customers less than 45% are satisfied with the service and prices, we need to use the Binomial Distribution.
The Binomial Distribution is used when there are only two possible outcomes, success or failure, and each outcome has a fixed probability of occurring.
In this case, the success is defined as a customer being satisfied with the service and prices, and the failure is a customer being dissatisfied.
Let's break down the information provided:
- The bookstore claims that 50% of its customers are satisfied with the service and prices, which means the probability of success (p) is 0.50.
- We want to find the probability that less than 45% of a random sample of 600 customers are satisfied. In other words, we want to find P(X < 0.45 * 600), where X is the random variable representing the number of satisfied customers in the sample.
To calculate this probability, we can use the cumulative distribution function (CDF) of the Binomial Distribution formula:
P(X < k) = Σ from r = 0 to k - 1 (nCᵣ) * pᵣ * (1 - p)^(n - r)
Where:
- n is the total number of trials (600 in this case),
- k is the number of successful outcomes we want (0.45 * 600 = 270),
- p is the probability of success (0.50), and
- nCᵣ (read as "n choose r") is the binomial coefficient and represents the number of combinations of n items taken r at a time.
To calculate P(X < k), we need to sum up the probabilities for all values r from 0 to k - 1. In this case, k = 270.
Computing this sum can be time-consuming and complicated. However, with the help of statistical software or online calculators, you can easily obtain the probability.