Facebook reports that the average number of Facebook friends worldwide is 175.5 with a standard deviation of 90.57. If you were to take a sample of 30 students, what is the probability that the mean number Facebook friends in the sample will be 190 friends or more?

To solve this problem, we will use the Central Limit Theorem (CLT) and the z-score to find the probability.

Step 1: Determine the population parameters.
- The population mean (μ) is 175.5.
- The population standard deviation (σ) is 90.57.

Step 2: Calculate the standard error of the sample mean.
The standard error (σₘ) represents the standard deviation of the sample means distribution and is calculated using: σₘ = σ / √n.
- σ is the population standard deviation, which is 90.57.
- n is the sample size, which is 30 in this case.

σₘ = 90.57 / √30
σₘ ≈ 16.52

Step 3: Convert the given value into a z-score.
To find the z-score, we use the formula: z = (x - μ) / σₘ.
- x is the value we are interested in (190 friends).
- μ is the population mean (175.5).
- σₘ is the standard error of the sample mean (16.52).

z = (190 - 175.5) / 16.52
z ≈ 0.878

Step 4: Find the probability using the z-table.
We need to find the probability of getting a z-score greater than or equal to 0.878 (P(z ≥ 0.878)). This represents the probability of getting a sample mean of 190 friends or more.

Using the z-table, P(z ≥ 0.878) is approximately 0.1906.

Step 5: Interpret the result.
The probability that the mean number of Facebook friends in the sample of 30 students will be 190 friends or more is approximately 0.1906, or 19.06%.

Note: This calculation assumes that the sample is selected randomly and the sample size is large enough for the CLT to apply.