Question

quiz
find the probability for each. (assume this is a standard normal distribution.)
a. ( p(0 < z < 2.53) )
b. ( p(z < 1.73) )
c. ( p(z > 1.98) )

Explanation:

To find the probabilities for the standard normal distribution, we use the Z - table (standard normal table) which gives the area to the left of a given Z - score.

Part a: \(P(0 < z < 2.53)\)

Step 1: Recall the property of the standard normal distribution

The total area under the standard normal curve is 1, and the curve is symmetric about \(z = 0\). The area to the left of \(z=0\) is \(0.5\) (since \(P(z < 0)=0.5\) for a standard normal distribution).
We need to find \(P(0 < z < 2.53)=P(z < 2.53)-P(z < 0)\)

Step 2: Find \(P(z < 2.53)\) from the Z - table

Looking up \(z = 2.53\) in the standard normal table, we find that \(P(z < 2.53)=0.9943\)

Step 3: Calculate \(P(0 < z < 2.53)\)

We know that \(P(z < 0) = 0.5\)
So, \(P(0 < z < 2.53)=0.9943 - 0.5=0.4943\)

Part b: \(P(z < 1.73)\)

Step 1: Use the Z - table

We directly look up \(z = 1.73\) in the standard normal table.

Step 2: Find the value from the Z - table

Looking up \(z = 1.73\) in the Z - table, we get \(P(z < 1.73)=0.9582\)

Part c: \(P(z>1.98)\)

Step 1: Recall the property of the standard normal distribution

We know that \(P(z > a)=1 - P(z < a)\)

Step 2: Find \(P(z < 1.98)\) from the Z - table

Looking up \(z = 1.98\) in the Z - table, we find that \(P(z < 1.98)=0.9761\)

Step 3: Calculate \(P(z > 1.98)\)

Using the formula \(P(z > 1.98)=1 - P(z < 1.98)\)
Substitute \(P(z < 1.98) = 0.9761\) into the formula: \(P(z > 1.98)=1 - 0.9761 = 0.0239\)

Answer:

a. \(P(0 < z < 2.53)=\boldsymbol{0.4943}\)

b. \(P(z < 1.73)=\boldsymbol{0.9582}\)

c. \(P(z > 1.98)=\boldsymbol{0.0239}\)