Respuesta :
Answer:
a) 0.4052; b) 0.0587; c) 0.7389; d) 0.3156
Step-by-step explanation:
We use z scores to find these probabilities. The formula for the z score of an individual value is
[tex]z=\frac{X-\mu}{\sigma}[/tex]
Our mean, μ, is 2.4 and our standard deviation, σ, is 10.
For part a,
We want P(X < 0). Plugging this value into our formula for a z score, we have
z = (0-2.4)/10 = -2.4/10 = -0.24
Using a z table, we see that the area under the curve to the left of this value is 0.4052. This is the probability x is less than 0.
For part b,
We want P(-19 < X < -12). We find the z score of both endpoints and then subtract their areas:
z = (-19-2.4)/10 = -21.4/10 = -2.14
z = (-12-2.4)/10 = -14.4/10 = -1.44
The area under the curve to the left of z = -2.14 is 0.0749. The area under the curve to the left of z = -1.44 is 0.0162. This means the area between them is
0.0749-0.0162 = 0.0587.
For part c,
We want P(X > -4). Since the z table gives us the area under the curve to the left of each value, we will find P(X < -4) and then subtract that from 1:
z = (-4-2.4)/10 = -6.4/10 = -0.64
The area under the curve tot he left of this is 0.2611. This makes P(X > -4)
1 - 0.2611 = 0.7389.
For part d,
We will use the formula for a z-score of a sample mean:
[tex]z = \frac{X-\mu}{\sigma\div \sqrt{n}}[/tex]
We have the same mean and standard deviation as in all the other parts. The sample size is 4.
For P(X < 0),
z = (0-2.4)/(10÷√4) = -2.4/(10÷2) = -2.4/5 = -0.48
The area under the curve to the left of this is 0.3156.
