A well known social media company is looking to expand their online presence by creating another platform. They know that they current average 2 , 500 , 000 users each day, with a standard deviation of 625 , 000 users. If they randomly sample 50 days to analyze the use of their existing technology, identify each of the following, rounding to the nearest whole number if necessary. Do not round until the end of the calculation, do not include commas in your answers. Provide your answer below:

To solve this problem, we will first find the mean and standard deviation for the sample of 50 days, and then calculate the expected range of users for this sample.

1. Mean for the sample of 50 days:

Since the company currently averages 2,500,000 users each day with a standard deviation of 625,000 users, we can find the mean for the sample of 50 days as follows:

Mean (sample) = Mean (population) * Number of days

Mean (sample) = 2,500,000 * 50

Mean (sample) = 125,000,000

2. Standard deviation for the sample of 50 days:

To find the standard deviation for the sample, we will use the formula for the standard deviation of a sample:

Standard Deviation (sample) = Standard Deviation (population) / sqrt(Number of days)

Standard Deviation (sample) = 625,000 / sqrt(50)

Standard Deviation (sample) ≈ 625,000 / 7.071

Standard Deviation (sample) ≈ 88,975

3. Expected range of users for the sample of 50 days:

Now, we will find the range of users for the sample of 50 days by using the mean and standard deviation we calculated earlier. We will use the empirical rule, which states that for a normal distribution, approximately 99.7% of the data falls within 3 standard deviations of the mean.

Lower limit = Mean (sample) - 3 * Standard Deviation (sample)

Lower limit = 125,000,000 - 3 * 88,975

Lower limit ≈ 125,000,000 - 266,925

Lower limit ≈ 124,733,075

Upper limit = Mean (sample) + 3 * Standard Deviation (sample)

Upper limit = 125,000,000 + 3 * 88,975

Upper limit ≈ 125,000,000 + 266,925

Upper limit ≈ 125,266,925

So, the expected range of users for the sample of 50 days is approximately between 124,733,075 and 125,266,925 users.