To find the mean and standard deviation of sleep time for the population, we can use the z-score formula.
Let's assume the mean sleep time is μ, and the standard deviation is σ. We need to find the values of μ and σ.
Given that 5% of average sleep time is less than 9 hours, we can convert this value to a z-score using the z-score formula:
z = (x - μ) / σ,
where x is the value of 9 hours.
So, we have:
(9 - μ) / σ = -1.645
Similarly, given that 5% of average sleep time is more than 9.4 hours, we can convert this value to a z-score using the z-score formula:
z = (x - μ) / σ,
where x is the value of 9.4 hours.
So, we have:
(9.4 - μ) / σ = 1.645
Now, we have a system of two equations with two variables:
(9 - μ) / σ = -1.645
(9.4 - μ) / σ = 1.645
To solve this system, we can isolate μ in both equations:
9 - μ = -1.645σ
9.4 - μ = 1.645σ
Now, we can solve for μ by subtracting the first equation from the second equation:
(9.4 - μ) - (9 - μ) = 1.645σ + 1.645σ
0.4 = 3.29σ
Simplifying, we have:
σ = 0.4 / 3.29
Now, we substitute the value of σ back into either of the original equations to solve for μ. Let's choose the first equation:
(9 - μ) / (0.4 / 3.29) = -1.645
Multiplying both sides of the equation by (0.4 / 3.29) and simplifying, we get:
9 - μ = -0.2188
Subtracting -0.2188 from both sides of the equation, we get:
μ = 9 + 0.2188 = 9.2188
So, the mean sleep time for the population is approximately 9.2 hours (rounded to 1 decimal place).
The standard deviation is given by:
σ = 0.4 / 3.29 = 0.1217
So, the standard deviation of sleep time for the population is approximately 0.122 hours (rounded to 3 decimal places).