The time a group of high school students arrive home from school each day was found to be normally distributed. The mean time was 3:15pm and the times had a standard deviation of 15 minutes. What is the probability that a student chosen at random arrives home from school before 2:30pm?

what formula would i use to solve for this?

Z score = (x-μ)/SD, where x is the individual score, μ = mean, and SD = standard deviation.

Once the Z score is obtained, look it up in a table in the back of your statistics text called something like "areas under the normal distribution" to get the probability.

I hope this helps. Thanks for asking.

To solve this probability question, you can use the cumulative distribution function (CDF) of the normal distribution. The CDF calculates the probability that a random variable is less than or equal to a given value.

In this case, you want to find the probability that a student arrives home before 2:30pm (or 2:30pm in a 24-hour format). However, the mean and standard deviation are provided in a different format (3:15pm and 15 minutes). So, you need to convert the 2:30pm into the same format.

To do this, subtract the mean time (3:15pm) from the desired time (2:30pm) to get the difference in minutes. Convert this difference into minutes and then divide by the standard deviation (15 minutes). This will give you a standardized score (z-score) that can be used to find the probability using the standard normal distribution.

The formula to calculate the z-score is:
z = (x - μ) / σ

Where:
z is the standardized score
x is the desired time (in minutes)
μ is the mean time (in minutes)
σ is the standard deviation (in minutes)

After calculating the z-score, you can use the z-table or a statistical calculator to find the corresponding probability. The z-table gives you the area to the left of the z-score. Since you want the probability of arriving before 2:30pm, you need the area to the left of the z-score.

So, in summary, the formula you would use to solve for this is:
P(X < x) = Φ((x - μ) / σ)

Where:
P(X < x) is the probability that a student arrives home before the desired time
Φ is the cumulative distribution function (CDF) of the standard normal distribution
x is the desired time (in minutes)
μ is the mean time (in minutes)
σ is the standard deviation (in minutes)