Question

1. The lifetime of a certain type of automobile tire (in thousands of miles) is normally distributed with mean 38 and standard deviation 4 . Find the percentile corresponding to p = 67 % of the tire lifetimes. Write only a number as your answer. Round to two decimal places (for example: 0.81).

2. In a large Biology class, the scores of the final exam were approximately normally distributed with mean 80.5 and standard deviation 9.9. What proportion of final exam scores were greater than 71.6?
Write only a number as your answer. Round to 4 decimal places (for example 0.0048). Do not write as a percentage

3.

The average price for a bag of coffee in a certain city is normally distributed with mean 11.5 dollars and standard deviation 2.7 dollars. What proportion of coffee bags in this city cost less than 9.51 dollars?

Write only a number as your answer. Round to 4 decimal places (for example 0.0048). Do not write as a percentage.

Answer 1

1)

z value at 67% = 0.44

z = (x - mean)/s
0.44 =(x - 38)/4

x = 4 * 0.44 + 38
= 39.76

2)

Here, μ = 80.5, σ = 9.9 and x = 71.6. We need to compute P(X >= 71.6). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (71.6 - 80.5)/9.9 = -0.9

Therefore,
P(X >= 71.6) = P(z <= (71.6 - 80.5)/9.9)
= P(z >= -0.9)
= 1 - 0.1841 = 0.8159

3)

Here, μ = 11.5, σ = 2.7 and x = 9.51. We need to compute P(X <= 9.51). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (9.51 - 11.5)/2.7 = -0.74

Therefore,
P(X <= 9.51) = P(z <= (9.51 - 11.5)/2.7)
= P(z <= -0.74)
= 0.2296