Question

1) We know that z has a standard normal distribution with a mean of 0 and standard deviation of 1. The probability that z is less than 1.15 is  . Use your z-table and report your answer to four decimal places.

2)A sample of 15 grades from a recent Stats exam has a mean of 69.3 points (out of a possible 100 points) and a standard deviation of 16.5 points. Calculate the z-score for the student who scored 74.1 points on the exam.

3)The diameters of bolts produced by a certain machine are normally distributed with a population mean of 0.300 inches and a population standard deviation of 0.001 inches. What is the probability that a randomly selected bolt will have a diameter greater than 0.302 inches? Use 4 decimal places in answering.
4)The variable X is normally distributed. The population mean is 60.0, and the population standard deviation is 4.00. Find P(X < 53.0 ). Round your answer to four decimal places

5)The diameters of bolts produced by a certain machine are normally distributed with a population mean of 13.0 milimeters (mm) and a population standard deviation of 0.1 mm. Bolts with a diameter less than 12.822 must be discarded. What is the probability that a randomly selected bolt will have a diameter less than 12.822 mm? Use 4 decimal places in answering.

Answer 1

a)

= P(z <= 1.15)
= 0.8749

2)

z = (x - μ)/σ
z = (1.15 - 6.3)/4.2603 = -1.21

3)

Here, μ = 0.3, σ = 0.001 and x = 0.302. We need to compute P(X >= 0.302). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (0.302 - 0.3)/0.001 = 2

Therefore,
P(X >= 0.302) = P(z <= (0.302 - 0.3)/0.001)
= P(z >= 2)
= 1 - 0.9772 = 0.0228

4)

Here, μ = 60, σ = 4 and x = 53. We need to compute P(X <= 53). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (53 - 60)/4 = -1.75

Therefore,
P(X <= 53) = P(z <= (53 - 60)/4)
= P(z <= -1.75)
= 0.0401

5)

Here, μ = 13, σ = 0.1 and x = 12.822. We need to compute P(X <= 12.822). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (12.822 - 13)/0.1 = -1.78

Therefore,
P(X <= 12.822) = P(z <= (12.822 - 13)/0.1)
= P(z <= -1.78)
= 0.0375