A sample of size 50 will be drawn from a population with mean 12 and standard deviation 8 . Use the TI-84 calculator.
- Find the probability that x will be more than 10 . Round the answer to at least four decimal places.
- Find the 45th percentile of μ. Round the answer to two decimal places.
Answers
The T1-84 calculator answer are
- 0.9614
- 11.84
However, without using the T1-84 calculator, the answers are
- 0.9616
- 11.87
Step-by-step explanation:
Given that
Sample size = n = 50
Population mean = μ = 12
Standard Deviation = σ = 8
a. Here, we're to calculate the probability that the sample mean is greater than 10.
i.e P(X>10)
To do this, we'll calculate z - value using the following formula.
Z = (x - μ)/(σ/√n)
By substituton
Z = (10 - 12)/(8/√50)
Z = -1.77 ----- Approximately
The value of P(X > 10) is calculated by P(Z > -1.77)
So, P(X > 10) = P( Z > -1.77)
P(X>10) = P(Z<1.77)
P(X>10) = 0.961636
P(X>10) = 0.9616 ---- Approximated
b. Here, we're to calculate the 45th percentile of the population mean.
To do that, we'll make use of this formula
X=μ + Z, and we will use the standard normal distribution table.
Where μ = Population mean = 12
Z = the corresponding z score of 45th percentile
From z table, 45th percentile = -0.126
By substituton
X = μ + Zσ becomes
X = 12 + (-0.126*1)
X = 11.874
X = 11.87 --- Approximately