Gani, a regular coffee drinker, goes to his favorite coffee shop every night for a cup of hot cappuccino. The amount of time that he stays in his favorite spot is normally distributed with a mean of 45 minutes and a standard deviation of 5 minutes. Above how many minutes are the longest 10% of his stays in the coffee shop?

http://homepage.divms.uiowa.edu/~mbognar/applets/normal.html 51.4 minutes?

Solution pleaseeeee

Hey, try it and post your attempt.

Well, Gani must really love his coffee if he stays for such a long time! To answer your question, we need to use the concept of z-scores. The z-score tells us how many standard deviations a value is away from the mean.

To find the value above which the longest 10% of his stays fall, we need to find the z-score corresponding to the cutoff point. The z-score can be found using a standard normal distribution table or by using a calculator.

The z-score for a percentile can be found using the formula:
(z-score) = (value - mean) / standard deviation

For the longest 10% of his stays, we're looking for the cutoff point where only 10% of the data is above that point. The z-score for the 10th percentile is approximately -1.28.

Now, we can solve for the value using the formula:
(value) = (z-score * standard deviation) + mean

Substituting the values, we get:
(value) = (-1.28 * 5) + 45

Calculating this, we find:
(value) ≈ 38.6

So, the longest 10% of Gani's stays would be above approximately 38.6 minutes. That's quite a long time to savor a cappuccino!

To find the amount of time above which the longest 10% of Gani's stays in the coffee shop occur, we need to calculate the corresponding z-score and use it to find the corresponding value on the standard normal distribution.

Here's how you can do it step by step:

1. Calculate the z-score corresponding to the desired percentile (10% or 0.10). The z-score formula is:
z = (x - μ) / σ
where x is the value we want to find, μ is the mean, and σ is the standard deviation.

Substituting the known values:
z = (x - 45) / 5

2. To find the value x on the standard normal distribution, we need to find the z-score that corresponds to the desired percentile. We can use a standard normal distribution table or a calculator to find this value. Looking up the z-score of 0.10, we find that it is approximately -1.28.

3. Now, we can solve the equation for x:
-1.28 = (x - 45) / 5

Rearranging the equation to solve for x:
x - 45 = -1.28 * 5
x - 45 = -6.4
x = -6.4 + 45
x ≈ 38.6

Therefore, above approximately 38.6 minutes, Gani's stays in the coffee shop would be longer than the longest 10% of his stays.