91Ó°ÊÓ

For a continuous probability distribution, explain why the following holds true. $$ P(a

How do the width and height of a normal distribution change when its mean remains the same but its standard deviation decreases?

Do the width and/or height of a normal distribution change when its standard deviation remains the same but its mean increases?

For the standard normal distribution, find the area within one standard deviation of the mean - that is, the area between \(\mu-\sigma\) and \(\mu+\sigma .\)

Find the area under the standard normal curve a. between \(z=0\) and \(z=1.95\) b. between \(z=0\) and \(z=-2.05\) c. between \(z=1.15\) and \(z=2.37\) d. from \(z=-1.53\) to \(z=-2.88\) e. from \(z=-1.67\) to \(z=2.24\)

Let \(x\) denote the time taken to run a road race. Suppose \(x\) is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race a. in less than 160 minutes? b. in 215 to 245 minutes?

A construction zone on a highway has a posted speed limit of 40 miles per hour. The speeds of vehicles passing through this construction zone are normally distributed with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are a. exceeding the posted speed limit b. traveling at speeds between 50 and 57 miles per hour

According to the records of an electric company serving the Boston area, the mean electricity consumption for all households during winter is 1650 kilowatt-hours per month. Assume that the monthly electricity consumptions during winter by all households in this area have a normal distribution with a mean of 1650 kilowatt-hours and a standard deviation of 320 kilowatt-hours. a. Find the probability that the monthly electricity consumption during winter by a randomly selected household from this area is less than 1950 kilowatt- hours. b. What percentage of the households in this area have a monthly electricity consumption of 900 to 1300 kilowatt-hours?

The average monthly mortgage payment for all homeowners in a city is $$\$ 2850.$$ Suppose that the distribution of monthly mortgages paid by homeowners in this city follow an approximate normal distribution with a mean of $$\$ 2850$$ and a standard deviation of $$\$ 420.$$ Find the probability that the monthly mortgage paid by a randomly selected homeowner from this city is a. less than $$\$ 1200$$ b. between $$\$ 2300$$ and $$\$ 3140$$ c. more than $$\$ 3600$$ d. between $$\$ 3200$$ and $$\$ 3700$$

The management of a supermarket wants to adopt a new promotional policy of giving a free gift to every customer who spends more than a certain amount per visit at this supermarket. The expectation of the management is that after this promotional policy is advertised, the expenditures for all customers at this supermarket will be normally distributed with a mean of $$\$ 95$$ and a standard deviation of $$\$ 20.$$ If the management wants to give free gifts to at most \(10 \%\) of the customers, what should the amount be above which a customer would receive a free gift?