Final answer:
The probability that a randomly selected group of 50 shoppers will spend a total of more than $5100 can be calculated using the central limit theorem. The probability is very high.
Step-by-step explanation:
To find the probability that a randomly selected group of 50 shoppers will spend a total of more than $5100, we need to use the central limit theorem. The central limit theorem states that the distribution of sample means approaches a normal distribution as the sample size increases.
First, we calculate the mean of the sample means, which is equal to the mean of the population, $100. Next, we calculate the standard deviation of the sample means, which is equal to the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the sample means is $30/sqrt(50) = $4.24.
Using these values, we can convert the given total amount of $5100 into a z-score, which measures the number of standard deviations a value is away from the mean. The z-score is calculated as (x - mean) / standard deviation, where x is the given total amount. So, the z-score for $5100 is (5100 - 100) / 4.24 = 1200.94.
Finally, we can use a standard normal distribution table or a calculator to find the probability of a z-score being greater than 1200.94. The probability is very close to 1, which means that there is a very high chance that a randomly selected group of 50 shoppers will spend a total of more than $5100.