Final answer:
A list of five numbers where the mean is larger than all but one of the numbers is 1, 1, 1, 1, and 7. The mean of these numbers is 2.2, which is only smaller than the number 7. Use a calculator or computer to verify calculations and find standard deviations.
Step-by-step explanation:
To find a list of five numbers with a mean that is larger than all but one of the numbers, we can choose four numbers that are relatively lower and one number that is significantly higher to raise the average. Let's say we pick 1, 1, 1, 1 for the first four numbers since at least 25 percent of the values are equal to one, and as the fifth number, we pick a number greater than five but less than or equal to seven to satisfy the condition that the top 25 percent of the values fall between five and seven, inclusive. A possible fifth number to achieve our goal could be 7.
Our list of five numbers would be 1, 1, 1, 1, and 7. The sum of these numbers is 1+1+1+1+7=11. The mean of these numbers is 11/5=2.2. Since 7 is the only number in the list that is greater than 2.2, the mean is indeed larger than all but one of the numbers.
To confirm calculations and find standard deviations in different contexts, you can use your calculator or computer. For example, if we want to find a value that is two standard deviations above the mean, we first calculate the mean and standard deviation of the dataset and then add twice the standard deviation to the mean value.