Question

UCF students pay an average of 1,229 dollars for books. The standard deviation of costs of books is sigma =102 dollars. Select 36 UCF students at random. Find the probability that the mean cost of books of 36 students exceeds 1,229+(1.96)(102)/6.

Answer 1

The probability of the mean cost of books for 36 students exceeding $1,229 + (1.96)(102)/6 is calculated using the central limit theorem. The Z-score is found to be significantly high, meaning the probability is extremely close to zero.

Step-by-step explanation:

A student asked what the probability is that the mean cost of books for 36 randomly selected UCF students exceeds $1,229 + (1.96)(102)/6. To find this probability, we use the central limit theorem since we have a sufficiently large sample size. The sampling distribution of the sample mean will be approximately normally distributed with a mean of $1,229 and a standard error of σ/√{n}, where σ is the population standard deviation and n is the sample size.

First, calculate the standard error: SE = σ/√{n} = 102/√{36} = 102/6 = 17.

Next, determine the z-score for the value $1,229 + (1.96)(102)/6 by subtracting the population mean, $1,229, and dividing by the standard error, 17.

Z = ($1,229 + (1.96)(102)/6) - $1,229)/17 = (1.96)(102)/6/17 = 1.96 × 17 = 33.32.

Since 33.32 is significantly high and Z-scores this high are not typical, the probability of the sample mean exceeding this value is extremely close to zero. Thus, the probability that the mean cost of books for 36 students exceeds $1,229 + (1.96)(102)/6 is very unlikely or almost zero.