Normal Distribution Practice

1. Given the distributions of the random variables as indicated, determine the following probabilities. Draw a sketch of the distribution for each.

(a) ; use the table

(b) ; use the table

(d) ; use the calculator

(e) ; use the calculator

2. (a) It was estimated that batting averages in major league baseball in the 1940s were distributed with a mean of 0.260 and a standard deviation of 0.07. What percentage of the batters would have an average greater 0.400?

(b) Recent estimates have the batting averages distributed with the same mean but less variability. The new standard deviation is estimated to be 0.05. What percentage of the players would bat over 0.400 today?

3. Equipment maintenance at a major factory operates on the principle of preventative maintenance to avoid a complete shutdown of the assembly line if a component fails. If one of the components has an average lifetime of 321 hours with a standard deviation of 23 hours, determine how frequently the component should be replaced so that the probability of it failing during operation is less than 0.001.

4. In a certain process, ball bearings are manufactured with a mean diameter of 2 mm and a standard deviation of 0.01 mm. Using the same process in space, the process yields a mean of 2 mm and a standard deviation of 0.001 mm. Bearings are rejected if they are less than 1.98 mm or greater than 2.02 mm in diameter. Assuming that the diameters are normally distributed, what are the rejection percentages on Earth and in space?

5. (a) Given: σ=25, P(X<68)=0.63. Find μ.

(b) Given μ=-15, P(X<-20)=0.15. Find σ.

6. On a certain section of highway, 90% of the motorists drive at speeds greater than 100 km/h, and only 5% of the motorists drive at speeds less than 95 km/h. If the speeds are assumed to be normally distributed,

(a) determine the mean speed and its standard deviation

(b) determine the percentage of drivers whose speed is greater than 120 km/h.