A popular online retailer sells a wide variety of products including books. The proportion of customers that order only books is 21%. If a random sample of 600 customers is taken, what is the probability that more than 24% order only books?
Use the normal approximation to the binomial distribution.
First find mean and standard deviation.
mean = np = (600)(.21) = ?
standard deviation = √npq = √(600)(.21)(.79) = ?
Note: q = 1 - p
I'll let you finish the calculations.
Next take .24 times 600. Use that value for x in the z-score equation:
z = (x - mean)/sd
Once you calculate the z-score, look at a z-table to determine the probability. Remember that the problem is asking "more than" when you check the table.
I hope this will help get you started.
0.983
To answer this question, we need to use the concept of the binomial distribution.
The binomial distribution is used when there are two outcomes (success or failure), and each outcome has a fixed probability of occurring. In this case, the two outcomes are customers ordering only books (success) or customers ordering something other than books (failure), and the fixed probability is 21%.
Let's calculate the probability of exactly 24% ordering only books first. The probability of success (ordering only books) is 21%, so the probability of failure (ordering something other than books) is 1 - 21% = 79%.
The probability mass function (PMF) of the binomial distribution is given by the formula:
P(X = k) = nCk * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of getting exactly k successes
- n is the total number of trials (sample size in this case)
- k is the number of successes (customers ordering only books in this case)
- p is the probability of success (21% or 0.21)
- (1 - p) is the probability of failure (79% or 0.79)
- nCk is the number of combinations of n items taken k at a time, given by the formula nCk = n! / (k! * (n - k)!)
Using this formula, we can calculate the probability of exactly 24% ordering only books.
P(X = 24% of 600) = 600C(0.24*600) * (0.21)^(0.24*600) * (1 - 0.21)^(600 - 0.24*600)
Next, we need to calculate the probability of more than 24% ordering only books.
P(X > 24% of 600) = 1 - P(X = 24% of 600)
You can use a calculator or statistical software to perform these calculations. For example, you can use a programming language like R or Python with the binomial distribution functions to calculate this probability.